title: Logistic regression

text: We wish to fit a logistic function to the data consisting of the hours studied ("x") and the outcome of the test ("y" =1 for pass, 0 for fail). The data points are indexed by the subscript "k" which runs from formula_1 to formula_2. The "x" variable is called the "explanatory variable", and the "y" variable is called the "categorical variable" consisting of two categories: "pass" or "fail" corresponding to the categorical values 1 and 0 respectively.


title: Logistic regression

text: where "μ" is a location parameter (the midpoint of the curve, where formula_4) and "s" is a scale parameter. This expression may be rewritten as:


title: Logistic regression

text: where formula_6 and is known as the intercept (it is the "vertical" intercept or "y"-intercept of the line formula_7), and formula_8 (inverse scale parameter or rate parameter): these are the "y"-intercept and slope of the log-odds as a function of "x". Conversely, formula_9 and formula_10.


title: Logistic regression

text: The usual measure of goodness of fit for a logistic regression uses logistic loss (or log loss), the negative log-likelihood. For a given "x" and "y", write formula_11. The are the probabilities that the corresponding will be unity and are the probabilities that they will be zero (see Bernoulli distribution). We wish to find the values of and which give the "best fit" to the data. In the case of linear regression, the sum of the squared deviations of the fit from the data points ("y"), the squared error loss, is taken as a measure of the goodness of fit, and the best fit is obtained when that function is "minimized".


title: Logistic regression

text: The log loss can be interpreted as the "surprisal" of the actual outcome relative to the prediction , and is a measure of information content. Note that log loss is always greater than or equal to 0, equals 0 only in case of a perfect prediction (i.e., when formula_13 and formula_14, or formula_15 and formula_16), and approaches infinity as the prediction gets worse (i.e., when formula_14 and formula_18 or formula_19 and formula_20), meaning the actual outcome is "more surprising". Since the value of the logistic function is always strictly between zero and one, the log loss is always greater than zero and less than infinity. Note that unlike in a linear regression, where the model can have zero loss at a point by passing through a data point (and zero loss overall if all points are on a line), in a logistic regression it is not possible to have zero loss at any points, since is either 0 or 1, but .


title: Logistic regression

text: This expression is more formally known as the cross entropy of the predicted distribution formula_22 from the actual distribution formula_23, as probability distributions on the two-element space of (pass, fail).


title: Logistic regression

text: The sum of these, the total loss, is the overall negative log-likelihood , and the best fit is obtained for those choices of and for which is "minimized".


title: Logistic regression

text: Alternatively, instead of "minimizing" the loss, one can "maximize" its inverse, the (positive) log-likelihood:


title: Logistic regression

text: or equivalently maximize the likelihood function itself, which is the probability that the given data set is produced by a particular logistic function:


title: Logistic regression

text: Since "ℓ" is nonlinear in and , determining their optimum values will require numerical methods. Note that one method of maximizing "ℓ" is to require the derivatives of "ℓ" with respect to and to be zero:


title: Logistic regression

text: and the maximization procedure can be accomplished by solving the above two equations for and , which, again, will generally require the use of numerical methods.


title: Logistic regression

text: The and coefficients may be entered into the logistic regression equation to estimate the probability of passing the exam.


title: Logistic regression

text: For example, for a student who studies 2 hours, entering the value formula_32 into the equation gives the estimated probability of passing the exam of 0.25:


title: Logistic regression

text: By the Wald test, the output indicates that hours studying is significantly associated with the probability of passing the exam (formula_37). Rather than the Wald method, the recommended method to calculate the "p"-value for logistic regression is the likelihood-ratio test (LRT), which for this data gives formula_38 (see below).


title: Logistic regression

text: This simple model is an example of binary logistic regression, and has one explanatory variable and a binary categorical variable which can assume one of two categorical values. Multinomial logistic regression is the generalization of binary logistic regression to include any number of explanatory variables and any number of categories.


title: Logistic regression

text: An explanation of logistic regression can begin with an explanation of the standard logistic function. The logistic function is a sigmoid function, which takes any real input formula_39, and outputs a value between zero and one. For the logit, this is interpreted as taking input log-odds and having output probability. The "standard" logistic function formula_40 is defined as follows:


title: Logistic regression

text: Let us assume that formula_39 is a linear function of a single explanatory variable formula_43 (the case where formula_39 is a "linear combination" of multiple explanatory variables is treated similarly). We can then express formula_39 as follows:


title: Logistic regression

text: In the logistic model, formula_49 is interpreted as the probability of the dependent variable formula_50 equaling a success/case rather than a failure/non-case. It's clear that the response variables formula_51 are not identically distributed: formula_52 differs from one data point formula_53 to another, though they are independent given design matrix formula_54 and shared parameters formula_55.


title: Logistic regression

text: We can now define the logit (log odds) function as the inverse formula_56 of the standard logistic function. It is easy to see that it satisfies:


title: Logistic regression

text: The odds of the dependent variable equaling a case (given some linear combination formula_43 of the predictors) is equivalent to the exponential function of the linear regression expression. This illustrates how the logit serves as a link function between the probability and the linear regression expression. Given that the logit ranges between negative and positive infinity, it provides an adequate criterion upon which to conduct linear regression and the logit is easily converted back into the odds.


title: Logistic regression

text: So we define odds of the dependent variable equaling a case (given some linear combination formula_43 of the predictors) as follows:


title: Logistic regression

text: This exponential relationship provides an interpretation for formula_72: The odds multiply by formula_73 for every 1-unit increase in x.


title: Logistic regression

text: For a binary independent variable the odds ratio is defined as formula_74 where "a", "b", "c" and "d" are cells in a 2×2 contingency table.


title: Logistic regression

text: If there are multiple explanatory variables, the above expression formula_75 can be revised to formula_76. Then when this is used in the equation relating the log odds of a success to the values of the predictors, the linear regression will be a multiple regression with "m" explanators; the parameters formula_77 for all formula_78 are all estimated.


title: Logistic regression

text: The basic setup of logistic regression is as follows. We are given a dataset containing "N" points. Each point "i" consists of a set of "m" input variables "x" ... "x" (also called independent variables, explanatory variables, predictor variables, features, or attributes), and a binary outcome variable "Y" (also known as a dependent variable, response variable, output variable, or class), i.e. it can assume only the two possible values 0 (often meaning "no" or "failure") or 1 (often meaning "yes" or "success"). The goal of logistic regression is to use the dataset to create a predictive model of the outcome variable.


title: Logistic regression

text: As in linear regression, the outcome variables "Y" are assumed to depend on the explanatory variables "x" ... "x".


title: Logistic regression

text: The explanatory variables may be of any type: real-valued, binary, categorical, etc. The main distinction is between continuous variables and discrete variables.


title: Logistic regression

text: Formally, the outcomes "Y" are described as being Bernoulli-distributed data, where each outcome is determined by an unobserved probability "p" that is specific to the outcome at hand, but related to the explanatory variables. This can be expressed in any of the following equivalent forms:


title: Logistic regression

text: The basic idea of logistic regression is to use the mechanism already developed for linear regression by modeling the probability "p" using a linear predictor function, i.e. a linear combination of the explanatory variables and a set of regression coefficients that are specific to the model at hand but the same for all trials. The linear predictor function formula_83 for a particular data point "i" is written as:


title: Logistic regression

text: where formula_85 are regression coefficients indicating the relative effect of a particular explanatory variable on the outcome.


title: Logistic regression

text: The above example of binary logistic regression on one explanatory variable can be generalized to binary logistic regression on any number of explanatory variables "x, x..." and any number of categorical values formula_87.


title: Logistic regression

text: To begin with, we may consider a logistic model with "M" explanatory variables, "x", "x" ... "x" and, as in the example above, two categorical values ("y" = 0 and 1). For the simple binary logistic regression model, we assumed a linear relationship between the predictor variable and the log-odds (also called logit) of the event that formula_88. This linear relationship may be extended to the case of "M" explanatory variables:


title: Logistic regression

text: where "t" is the log-odds and formula_90 are parameters of the model. An additional generalization has been introduced in which the base of the model ("b") is not restricted to the Euler number "e". In most applications, the base formula_91 of the logarithm is usually taken to be "e". However, in some cases it can be easier to communicate results by working in base 2 or base 10.


title: Logistic regression

text: For a more compact notation, we will specify the explanatory variables and the "β" coefficients as -dimensional vectors:


title: Logistic regression

text: where formula_97 is the sigmoid function with base formula_91. The above formula shows that once the formula_99 are fixed, we can easily compute either the log-odds that formula_88 for a given observation, or the probability that formula_88 for a given observation. The main use-case of a logistic model is to be given an observation x, and estimate the probability "p(x)" that formula_88. The optimum beta coefficients may again be found by maximizing the log-likelihood. For "K" measurements, defining x as the explanatory vector of the "k"-th measurement, and "y" as the categorical outcome of that measurement, the log likelihood may be written in a form very similar to the simple formula_103 case above:


title: Logistic regression

text: As in the simple example above, finding the optimum "β" parameters will require numerical methods. One useful technique is to equate the derivatives of the log likelihood with respect to each of the "β" parameters to zero yielding a set of equations which will hold at the maximum of the log likelihood:


title: Logistic regression

text: Consider an example with formula_106 explanatory variables, formula_107, and coefficients formula_108, formula_109, and formula_110 which have been determined by the above method. To be concrete, the model is:


title: Logistic regression

text: In the above cases of two categories (binomial logistic regression), the categories were indexed by "0" and "1", and we had two probability distributions: The probability that the outcome was in category 1 was given by formula_139and the probability that the outcome was in category 0 was given by formula_140. The sum of both probabilities is equal to unity, as they must be.


title: Logistic regression

text: In general, if we have explanatory variables (including "x") and categories, we will need separate probability distributions, one for each category, indexed by "n", which describe the probability that the categorical outcome "y" for explanatory vector x will be in category "y=n". It will also be required that the sum of these probabilities over all categories be equal to unity. Using the mathematically convenient base "e", these probabilities are:


title: Logistic regression

text: Each of the probabilities except formula_144 will have their own set of regression coefficients formula_145. It can be seen that, as required, the sum of the formula_146 over all categories is unity. Note that the selection of formula_144 to be defined in terms of the other probabilities is artificial. Any of the probabilities could have been selected to be so defined. This special value of "n" is termed the "pivot index", and the log-odds ("t") are expressed in terms of the pivot probability and are again expressed as a linear combination of the explanatory variables:


title: Logistic regression

text: Note also that for the simple case of formula_149, the two-category case is recovered, with formula_150 and formula_151.


title: Logistic regression

text: The log-likelihood that a particular set of "K" measurements or data points will be generated by the above probabilities can now be calculated. Indexing each measurement by "k", let the "k"-th set of measured explanatory variables be denoted by formula_152 and their categorical outcomes be denoted by formula_153 which can be equal to any integer in [0,N]. The log-likelihood is then:


title: Logistic regression

text: where formula_155 is an indicator function which is equal to unity if "y = n" and zero otherwise. In the case of two explanatory variables, this indicator function was defined as "y" when "n" = 1 and "1-y" when "n" = 0. This was convenient, but not necessary. Again, the optimum beta coefficients may be found by maximizing the log-likelihood function generally using numerical methods. A possible method of solution is to set the derivatives of the log-likelihood with respect to each beta coefficient equal to zero and solve for the beta coefficients:


title: Logistic regression

text: where formula_157 is the "m"-th coefficient of the formula_145 vector and formula_159 is the "m"-th explanatory variable of the "k"-th measurement. Once the beta coefficients have been estimated from the data, we will be able to estimate the probability that any subsequent set of explanatory variables will result in any of the possible outcome categories.


title: Logistic regression

text: There are various equivalent specifications and interpretations of logistic regression, which fit into different types of more general models, and allow different generalizations.


title: Logistic regression

text: The particular model used by logistic regression, which distinguishes it from standard linear regression and from other types of regression analysis used for binary-valued outcomes, is the way the probability of a particular outcome is linked to the linear predictor function:


title: Logistic regression

text: This formulation expresses logistic regression as a type of generalized linear model, which predicts variables with various types of probability distributions by fitting a linear predictor function of the above form to some sort of arbitrary transformation of the expected value of the variable.


title: Logistic regression

text: The intuition for transforming using the logit function (the natural log of the odds) was explained above. It also has the practical effect of converting the probability (which is bounded to be between 0 and 1) to a variable that ranges over formula_162 — thereby matching the potential range of the linear prediction function on the right side of the equation.


title: Logistic regression

text: Note that both the probabilities "p" and the regression coefficients are unobserved, and the means of determining them is not part of the model itself. They are typically determined by some sort of optimization procedure, e.g. maximum likelihood estimation, that finds values that best fit the observed data (i.e. that give the most accurate predictions for the data already observed), usually subject to regularization conditions that seek to exclude unlikely values, e.g. extremely large values for any of the regression coefficients. The use of a regularization condition is equivalent to doing maximum a posteriori (MAP) estimation, an extension of maximum likelihood. (Regularization is most commonly done using a squared regularizing function, which is equivalent to placing a zero-mean Gaussian prior distribution on the coefficients, but other regularizers are also possible.) Whether or not regularization is used, it is usually not possible to find a closed-form solution; instead, an iterative numerical method must be used, such as iteratively reweighted least squares (IRLS) or, more commonly these days, a quasi-Newton method such as the L-BFGS method.


title: Logistic regression

text: The interpretation of the "β" parameter estimates is as the additive effect on the log of the odds for a unit change in the "j" the explanatory variable. In the case of a dichotomous explanatory variable, for instance, gender formula_163 is the estimate of the odds of having the outcome for, say, males compared with females.


title: Logistic regression

text: The formula can also be written as a probability distribution (specifically, using a probability mass function):


title: Logistic regression

text: The logistic model has an equivalent formulation as a latent-variable model. This formulation is common in the theory of discrete choice models and makes it easier to extend to certain more complicated models with multiple, correlated choices, as well as to compare logistic regression to the closely related probit model.


title: Logistic regression

text: Imagine that, for each trial "i", there is a continuous latent variable "Y" (i.e. an unobserved random variable) that is distributed as follows:


title: Logistic regression

text: i.e. the latent variable can be written directly in terms of the linear predictor function and an additive random error variable that is distributed according to a standard logistic distribution.


title: Logistic regression

text: The choice of modeling the error variable specifically with a standard logistic distribution, rather than a general logistic distribution with the location and scale set to arbitrary values, seems restrictive, but in fact, it is not. It must be kept in mind that we can choose the regression coefficients ourselves, and very often can use them to offset changes in the parameters of the error variable's distribution. For example, a logistic error-variable distribution with a non-zero location parameter "μ" (which sets the mean) is equivalent to a distribution with a zero location parameter, where "μ" has been added to the intercept coefficient. Both situations produce the same value for "Y" regardless of settings of explanatory variables. Similarly, an arbitrary scale parameter "s" is equivalent to setting the scale parameter to 1 and then dividing all regression coefficients by "s". In the latter case, the resulting value of "Y" will be smaller by a factor of "s" than in the former case, for all sets of explanatory variables — but critically, it will always remain on the same side of 0, and hence lead to the same "Y" choice.


title: Logistic regression

text: It turns out that this formulation is exactly equivalent to the preceding one, phrased in terms of the generalized linear model and without any latent variables. This can be shown as follows, using the fact that the cumulative distribution function (CDF) of the standard logistic distribution is the logistic function, which is the inverse of the logit function, i.e.


title: Logistic regression

text: This formulation—which is standard in discrete choice models—makes clear the relationship between logistic regression (the "logit model") and the probit model, which uses an error variable distributed according to a standard normal distribution instead of a standard logistic distribution. Both the logistic and normal distributions are symmetric with a basic unimodal, "bell curve" shape. The only difference is that the logistic distribution has somewhat heavier tails, which means that it is less sensitive to outlying data (and hence somewhat more robust to model mis-specifications or erroneous data).


title: Logistic regression

text: This model has a separate latent variable and a separate set of regression coefficients for each possible outcome of the dependent variable. The reason for this separation is that it makes it easy to extend logistic regression to multi-outcome categorical variables, as in the multinomial logit model. In such a model, it is natural to model each possible outcome using a different set of regression coefficients. It is also possible to motivate each of the separate latent variables as the theoretical utility associated with making the associated choice, and thus motivate logistic regression in terms of utility theory. (In terms of utility theory, a rational actor always chooses the choice with the greatest associated utility.) This is the approach taken by economists when formulating discrete choice models, because it both provides a theoretically strong foundation and facilitates intuitions about the model, which in turn makes it easy to consider various sorts of extensions. (See the example below.)


title: Logistic regression

text: The choice of the type-1 extreme value distribution seems fairly arbitrary, but it makes the mathematics work out, and it may be possible to justify its use through rational choice theory.


title: Logistic regression

text: It turns out that this model is equivalent to the previous model, although this seems non-obvious, since there are now two sets of regression coefficients and error variables, and the error variables have a different distribution. In fact, this model reduces directly to the previous one with the following substitutions:


title: Logistic regression

text: An intuition for this comes from the fact that, since we choose based on the maximum of two values, only their difference matters, not the exact values — and this effectively removes one degree of freedom. Another critical fact is that the difference of two type-1 extreme-value-distributed variables is a logistic distribution, i.e. formula_177 We can demonstrate the equivalent as follows:


title: Logistic regression

text: As an example, consider a province-level election where the choice is between a right-of-center party, a left-of-center party, and a secessionist party (e.g. the Parti Québécois, which wants Quebec to secede from Canada). We would then use three latent variables, one for each choice. Then, in accordance with utility theory, we can then interpret the latent variables as expressing the utility that results from making each of the choices. We can also interpret the regression coefficients as indicating the strength that the associated factor (i.e. explanatory variable) has in contributing to the utility — or more correctly, the amount by which a unit change in an explanatory variable changes the utility of a given choice. A voter might expect that the right-of-center party would lower taxes, especially on rich people. This would give low-income people no benefit, i.e. no change in utility (since they usually don't pay taxes); would cause moderate benefit (i.e. somewhat more money, or moderate utility increase) for middle-incoming people; would cause significant benefits for high-income people. On the other hand, the left-of-center party might be expected to raise taxes and offset it with increased welfare and other assistance for the lower and middle classes. This would cause significant positive benefit to low-income people, perhaps a weak benefit to middle-income people, and significant negative benefit to high-income people. Finally, the secessionist party would take no direct actions on the economy, but simply secede. A low-income or middle-income voter might expect basically no clear utility gain or loss from this, but a high-income voter might expect negative utility since he/she is likely to own companies, which will have a harder time doing business in such an environment and probably lose money.


title: Logistic regression

text: Yet another formulation combines the two-way latent variable formulation above with the original formulation higher up without latent variables, and in the process provides a link to one of the standard formulations of the multinomial logit.


title: Logistic regression

text: Here, instead of writing the logit of the probabilities "p" as a linear predictor, we separate the linear predictor into two, one for each of the two outcomes:


title: Logistic regression

text: Two separate sets of regression coefficients have been introduced, just as in the two-way latent variable model, and the two equations appear a form that writes the logarithm of the associated probability as a linear predictor, with an extra term formula_180 at the end. This term, as it turns out, serves as the normalizing factor ensuring that the result is a distribution. This can be seen by exponentiating both sides:


title: Logistic regression

text: In this form it is clear that the purpose of "Z" is to ensure that the resulting distribution over "Y" is in fact a probability distribution, i.e. it sums to 1. This means that "Z" is simply the sum of all un-normalized probabilities, and by dividing each probability by "Z", the probabilities become "normalized". That is:


title: Logistic regression

text: This shows clearly how to generalize this formulation to more than two outcomes, as in multinomial logit.


title: Logistic regression

text: In order to prove that this is equivalent to the previous model, note that the above model is overspecified, in that formula_186 and formula_187 cannot be independently specified: rather formula_188 so knowing one automatically determines the other. As a result, the model is nonidentifiable, in that multiple combinations of β and β will produce the same probabilities for all possible explanatory variables. In fact, it can be seen that adding any constant vector to both of them will produce the same probabilities:


title: Logistic regression

text: As a result, we can simplify matters, and restore identifiability, by picking an arbitrary value for one of the two vectors. We choose to set formula_190 Then,


title: Logistic regression

text: which shows that this formulation is indeed equivalent to the previous formulation. (As in the two-way latent variable formulation, any settings where formula_175 will produce equivalent results.)


title: Logistic regression

text: Note that most treatments of the multinomial logit model start out either by extending the "log-linear" formulation presented here or the two-way latent variable formulation presented above, since both clearly show the way that the model could be extended to multi-way outcomes. In general, the presentation with latent variables is more common in econometrics and political science, where discrete choice models and utility theory reign, while the "log-linear" formulation here is more common in computer science, e.g. machine learning and natural language processing.


title: Logistic regression

text: This functional form is commonly called a single-layer perceptron or single-layer artificial neural network. A single-layer neural network computes a continuous output instead of a step function. The derivative of "p" with respect to "X" = ("x", ..., "x") is computed from the general form:


title: Logistic regression

text: where "f"("X") is an analytic function in "X". With this choice, the single-layer neural network is identical to the logistic regression model. This function has a continuous derivative, which allows it to be used in backpropagation. This function is also preferred because its derivative is easily calculated:


title: Logistic regression

text: A closely related model assumes that each "i" is associated not with a single Bernoulli trial but with "n" independent identically distributed trials, where the observation "Y" is the number of successes observed (the sum of the individual Bernoulli-distributed random variables), and hence follows a binomial distribution:


title: Logistic regression

text: The regression coefficients are usually estimated using maximum likelihood estimation. Unlike linear regression with normally distributed residuals, it is not possible to find a closed-form expression for the coefficient values that maximize the likelihood function, so that an iterative process must be used instead; for example Newton's method. This process begins with a tentative solution, revises it slightly to see if it can be improved, and repeats this revision until no more improvement is made, at which point the process is said to have converged.


title: Logistic regression

text: In some instances, the model may not reach convergence. Non-convergence of a model indicates that the coefficients are not meaningful because the iterative process was unable to find appropriate solutions. A failure to converge may occur for a number of reasons: having a large ratio of predictors to cases, multicollinearity, sparseness, or complete separation.


title: Logistic regression

text: Binary logistic regression (formula_201 or formula_202) can, for example, be calculated using "iteratively reweighted least squares" (IRLS), which is equivalent to maximizing the log-likelihood of a Bernoulli distributed process using Newton's method. If the problem is written in vector matrix form, with parameters formula_203, explanatory variables formula_204 and expected value of the Bernoulli distribution formula_205, the parameters formula_206 can be found using the following iterative algorithm:


title: Logistic regression

text: The regressor matrix and formula_211 the vector of response variables. More details can be found in the literature.


title: Logistic regression

text: In a Bayesian statistics context, prior distributions are normally placed on the regression coefficients, for example in the form of Gaussian distributions. There is no conjugate prior of the likelihood function in logistic regression. When Bayesian inference was performed analytically, this made the posterior distribution difficult to calculate except in very low dimensions. Now, though, automatic software such as OpenBUGS, JAGS, PyMC3, Stan or Turing.jl allows these posteriors to be computed using simulation, so lack of conjugacy is not a concern. However, when the sample size or the number of parameters is large, full Bayesian simulation can be slow, and people often use approximate methods such as variational Bayesian methods and expectation propagation.


title: Logistic regression

text: A widely used rule of thumb, the "one in ten rule", states that logistic regression models give stable values for the explanatory variables if based on a minimum of about 10 events per explanatory variable (EPV); where "event" denotes the cases belonging to the less frequent category in the dependent variable. Thus a study designed to use formula_212 explanatory variables for an event (e.g. myocardial infarction) expected to occur in a proportion formula_213 of participants in the study will require a total of formula_214 participants. However, there is considerable debate about the reliability of this rule, which is based on simulation studies and lacks a secure theoretical underpinning. According to some authors the rule is overly conservative in some circumstances, with the authors stating, "If we (somewhat subjectively) regard confidence interval coverage less than 93 percent, type I error greater than 7 percent, or relative bias greater than 15 percent as problematic, our results indicate that problems are fairly frequent with 2–4 EPV, uncommon with 5–9 EPV, and still observed with 10–16 EPV. The worst instances of each problem were not severe with 5–9 EPV and usually comparable to those with 10–16 EPV".


title: Logistic regression

text: Others have found results that are not consistent with the above, using different criteria. A useful criterion is whether the fitted model will be expected to achieve the same predictive discrimination in a new sample as it appeared to achieve in the model development sample. For that criterion, 20 events per candidate variable may be required. Also, one can argue that 96 observations are needed only to estimate the model's intercept precisely enough that the margin of error in predicted probabilities is ±0.1 with a 0.95 confidence level.


title: Logistic regression

text: In any fitting procedure, the addition of another fitting parameter to a model (e.g. the beta parameters in a logistic regression model) will almost always improve the ability of the model to predict the measured outcomes. This will be true even if the additional term has no predictive value, since the model will simply be "overfitting" to the noise in the data. The question arises as to whether the improvement gained by the addition of another fitting parameter is significant enough to recommend the inclusion of the additional term, or whether the improvement is simply that which may be expected from overfitting.


title: Logistic regression

text: In short, for logistic regression, a statistic known as the deviance is defined which is a measure of the error between the logistic model fit and the outcome data. In the limit of a large number of data points, the deviance is chi-squared distributed, which allows a chi-squared test to be implemented in order to determine the significance of the explanatory variables.


title: Logistic regression

text: Linear regression and logistic regression have many similarities. For example, in simple linear regression, a set of "K" data points ("x", "y") are fitted to a proposed model function of the form formula_215. The fit is obtained by choosing the "b" parameters which minimize the sum of the squares of the residuals (the squared error term) for each data point:


title: Logistic regression

text: The idea of a null model may be introduced, in which it is assumed that the "x" variable is of no use in predicting the y outcomes: The data points are fitted to a null model function of the form "y=b" with a squared error term:


title: Logistic regression

text: The fitting process consists of choosing a value of "b" which minimizes formula_219 of the fit to the null model, denoted by formula_220 where the formula_221 subscript denotes the null model. It is seen that the null model is optimized by formula_222 where formula_223 is the mean of the "y" values, and the optimized formula_220 is:


title: Logistic regression

text: which is proportional to the square of the (uncorrected) sample standard deviation of the "y" data points.


title: Logistic regression

text: We can imagine a case where the "y" data points are randomly assigned to the various "x", and then fitted using the proposed model. Specifically, we can consider the fits of the proposed model to every permutation of the "y" outcomes. It can be shown that the optimized error of any of these fits will never be less than the optimum error of the null model, and that the difference between these minimum error will follow a chi-squared distribution distribution, with degrees of freedom equal those of the proposed model minus those of the null model which, in this case, will be 2-1=1. Using the chi-squared test, we may then estimate how many of these permuted sets of "y" will yield an minimum error less than or equal to the minimum error using the original "y", and so we can estimate how significant an improvement is given by the inclusion of the "x" variable in the proposed model.


title: Logistic regression

text: For logistic regression, the measure of goodness-of-fit is the likelihood function "L", or its logarithm, the log-likelihood "ℓ". The likelihood function "L" is analogous to the formula_219 in the linear regression case, except that the likelihood is maximized rather than minimized. Denote the maximized log-likelihood of the proposed model by formula_227.


title: Logistic regression

text: In the case of simple binary logistic regression, the set of "K" data points are fitted in a probabilistic sense to a function of the form:


title: Logistic regression

text: where formula_223 is again the mean of the "y" values. Again, we can conceptually consider the fit of the proposed model to every permutation of the "y" and it can be shown that the maximum log-likelihood of these permutation fits will never be smaller than that of the null model:


title: Logistic regression

text: Also, as an analog to the error of the linear regression case, we may define the deviance of a logistic regression fit as:


title: Logistic regression

text: which will always be positive or zero. The reason for this choice is that not only is the deviance a good measure of the goodness of fit, it is also approximately chi-squared distributed, with the approximation improving as the number of data points ("K") increases, becoming exactly chi-square distributed in the limit of an infinite number of data points. As in the case of linear regression, we may use this fact to estimate the probability that a random set of data points will give a better fit than the fit obtained by the proposed model, and so have an estimate how significantly the model is improved by including the "x" data points in the proposed model.


title: Logistic regression

text: For the simple model of student test scores described above, the maximum value of the log-likelihood of the null model is formula_243 The maximum value of the log-likelihood for the simple model is formula_244 so that the deviance is formula_245


title: Logistic regression

text: Using the chi-squared test of significance, the integral of the chi-squared distribution with one degree of freedom from 11.6661... to infinity is equal to 0.00063649...


title: Logistic regression

text: This effectively means that about 6 out of a 10,000 fits to random "y" can be expected to have a better fit (smaller deviance) than the given "y" and so we can conclude that the inclusion of the "x" variable and data in the proposed model is a very significant improvement over the null model. In other words, we reject the null hypothesis with formula_246 confidence.


title: Logistic regression

text: Goodness of fit in linear regression models is generally measured using R. Since this has no direct analog in logistic regression, various methods including the following can be used instead.


title: Logistic regression

text: In linear regression analysis, one is concerned with partitioning variance via the sum of squares calculations – variance in the criterion is essentially divided into variance accounted for by the predictors and residual variance. In logistic regression analysis, deviance is used in lieu of a sum of squares calculations. Deviance is analogous to the sum of squares calculations in linear regression and is a measure of the lack of fit to the data in a logistic regression model. When a "saturated" model is available (a model with a theoretically perfect fit), deviance is calculated by comparing a given model with the saturated model. This computation gives the likelihood-ratio test:


title: Logistic regression

text: In the above equation, represents the deviance and ln represents the natural logarithm. The log of this likelihood ratio (the ratio of the fitted model to the saturated model) will produce a negative value, hence the need for a negative sign. can be shown to follow an approximate chi-squared distribution. Smaller values indicate better fit as the fitted model deviates less from the saturated model. When assessed upon a chi-square distribution, nonsignificant chi-square values indicate very little unexplained variance and thus, good model fit. Conversely, a significant chi-square value indicates that a significant amount of the variance is unexplained.


title: Logistic regression

text: When the saturated model is not available (a common case), deviance is calculated simply as −2·(log likelihood of the fitted model), and the reference to the saturated model's log likelihood can be removed from all that follows without harm.


title: Logistic regression

text: Two measures of deviance are particularly important in logistic regression: null deviance and model deviance. The null deviance represents the difference between a model with only the intercept (which means "no predictors") and the saturated model. The model deviance represents the difference between a model with at least one predictor and the saturated model. In this respect, the null model provides a baseline upon which to compare predictor models. Given that deviance is a measure of the difference between a given model and the saturated model, smaller values indicate better fit. Thus, to assess the contribution of a predictor or set of predictors, one can subtract the model deviance from the null deviance and assess the difference on a formula_248 chi-square distribution with degrees of freedom equal to the difference in the number of parameters estimated.


title: Logistic regression

text: If the model deviance is significantly smaller than the null deviance then one can conclude that the predictor or set of predictors significantly improve the model's fit. This is analogous to the -test used in linear regression analysis to assess the significance of prediction.


title: Logistic regression

text: In linear regression the squared multiple correlation, is used to assess goodness of fit as it represents the proportion of variance in the criterion that is explained by the predictors. In logistic regression analysis, there is no agreed upon analogous measure, but there are several competing measures each with limitations.


title: Logistic regression

text: The Hosmer–Lemeshow test uses a test statistic that asymptotically follows a formula_251 distribution to assess whether or not the observed event rates match expected event rates in subgroups of the model population. This test is considered to be obsolete by some statisticians because of its dependence on arbitrary binning of predicted probabilities and relative low power.


title: Logistic regression

text: After fitting the model, it is likely that researchers will want to examine the contribution of individual predictors. To do so, they will want to examine the regression coefficients. In linear regression, the regression coefficients represent the change in the criterion for each unit change in the predictor. In logistic regression, however, the regression coefficients represent the change in the logit for each unit change in the predictor. Given that the logit is not intuitive, researchers are likely to focus on a predictor's effect on the exponential function of the regression coefficient – the odds ratio (see definition). In linear regression, the significance of a regression coefficient is assessed by computing a "t" test. In logistic regression, there are several different tests designed to assess the significance of an individual predictor, most notably the likelihood ratio test and the Wald statistic.


title: Logistic regression

text: The likelihood-ratio test discussed above to assess model fit is also the recommended procedure to assess the contribution of individual "predictors" to a given model. In the case of a single predictor model, one simply compares the deviance of the predictor model with that of the null model on a chi-square distribution with a single degree of freedom. If the predictor model has significantly smaller deviance (c.f. chi-square using the difference in degrees of freedom of the two models), then one can conclude that there is a significant association between the "predictor" and the outcome. Although some common statistical packages (e.g. SPSS) do provide likelihood ratio test statistics, without this computationally intensive test it would be more difficult to assess the contribution of individual predictors in the multiple logistic regression case. To assess the contribution of individual predictors one can enter the predictors hierarchically, comparing each new model with the previous to determine the contribution of each predictor. There is some debate among statisticians about the appropriateness of so-called "stepwise" procedures. The fear is that they may not preserve nominal statistical properties and may become misleading.


title: Logistic regression

text: Alternatively, when assessing the contribution of individual predictors in a given model, one may examine the significance of the Wald statistic. The Wald statistic, analogous to the "t"-test in linear regression, is used to assess the significance of coefficients. The Wald statistic is the ratio of the square of the regression coefficient to the square of the standard error of the coefficient and is asymptotically distributed as a chi-square distribution.


title: Logistic regression

text: Although several statistical packages (e.g., SPSS, SAS) report the Wald statistic to assess the contribution of individual predictors, the Wald statistic has limitations. When the regression coefficient is large, the standard error of the regression coefficient also tends to be larger increasing the probability of Type-II error. The Wald statistic also tends to be biased when data are sparse.


title: Logistic regression

text: Suppose cases are rare. Then we might wish to sample them more frequently than their prevalence in the population. For example, suppose there is a disease that affects 1 person in 10,000 and to collect our data we need to do a complete physical. It may be too expensive to do thousands of physicals of healthy people in order to obtain data for only a few diseased individuals. Thus, we may evaluate more diseased individuals, perhaps all of the rare outcomes. This is also retrospective sampling, or equivalently it is called unbalanced data. As a rule of thumb, sampling controls at a rate of five times the number of cases will produce sufficient control data.


title: Logistic regression

text: Logistic regression is unique in that it may be estimated on unbalanced data, rather than randomly sampled data, and still yield correct coefficient estimates of the effects of each independent variable on the outcome. That is to say, if we form a logistic model from such data, if the model is correct in the general population, the formula_77 parameters are all correct except for formula_65. We can correct formula_65 if we know the true prevalence as follows:


title: Logistic regression

text: Like other forms of regression analysis, logistic regression makes use of one or more predictor variables that may be either continuous or categorical. Unlike ordinary linear regression, however, logistic regression is used for predicting dependent variables that take membership in one of a limited number of categories (treating the dependent variable in the binomial case as the outcome of a Bernoulli trial) rather than a continuous outcome. Given this difference, the assumptions of linear regression are violated. In particular, the residuals cannot be normally distributed. In addition, linear regression may make nonsensical predictions for a binary dependent variable. What is needed is a way to convert a binary variable into a continuous one that can take on any real value (negative or positive). To do that, binomial logistic regression first calculates the odds of the event happening for different levels of each independent variable, and then takes its logarithm to create a continuous criterion as a transformed version of the dependent variable. The logarithm of the odds is the of the probability, the is defined as follows:


title: Logistic regression

text: The of the probability of success is then fitted to the predictors. The predicted value of the is converted back into predicted odds, via the inverse of the natural logarithm – the exponential function. Thus, although the observed dependent variable in binary logistic regression is a 0-or-1 variable, the logistic regression estimates the odds, as a continuous variable, that the dependent variable is a 'success'. In some applications, the odds are all that is needed. In others, a specific yes-or-no prediction is needed for whether the dependent variable is or is not a 'success'; this categorical prediction can be based on the computed odds of success, with predicted odds above some chosen cutoff value being translated into a prediction of success.


title: Logistic regression

text: Of all the functional forms used for estimating the probabilities of a particular categorical outcome which optimize the fit by maximizing the likelihood function (e.g. probit regression, Poisson regression, etc.), the logistic regression solution is unique in that it is a maximum entropy solution. This is a case of a general property: an exponential family of distributions maximizes entropy, given an expected value. In the case of the logistic model, the logistic function is the natural parameter of the Bernoulli distribution (it is in "canonical form", and the logistic function is the canonical link function), while other sigmoid functions are non-canonical link functions; this underlies its mathematical elegance and ease of optimization. See for details.


title: Logistic regression

text: In order to show this, we use the method of Lagrange multipliers. The Lagrangian is equal to the entropy plus the sum of the products of Lagrange multipliers times various constraint expressions. The general multinomial case will be considered, since the proof is not made that much simpler by considering simpler cases. Equating the derivative of the Lagrangian with respect to the various probabilities to zero yields a functional form for those probabilities which corresponds to those used in logistic regression.


title: Logistic regression

text: As in the above section on , we will consider explanatory variables denoted and which include formula_260. There will be a total of "K" data points, indexed by formula_261, and the data points are given by formula_159 and . The "x" will also be represented as an -dimensional vector formula_263. There will be possible values of the categorical variable "y" ranging from 0 to N.


title: Logistic regression

text: Let "p(x)" be the probability, given explanatory variable vector x, that the outcome will be formula_264. Define formula_265 which is the probability that for the "k"-th measurement, the categorical outcome is "n".


title: Logistic regression

text: The Lagrangian will be expressed as a function of the probabilities "p" and will minimized by equating the derivatives of the Lagrangian with respect to these probabilities to zero. An important point is that the probabilities are treated equally and the fact that they sum to unity is part of the Lagrangian formulation, rather than being assumed from the beginning.


title: Logistic regression

text: Assuming the multinomial logistic function, the derivative of the log-likelihood with respect the beta coefficients was found to be:


title: Logistic regression

text: A very important point here is that this expression is (remarkably) not an explicit function of the beta coefficients. It is only a function of the probabilities "p" and the data. Rather than being specific to the assumed multinomial logistic case, it is taken to be a general statement of the condition at which the log-likelihood is maximized and makes no reference to the functional form of "p". There are then ("M"+1)("N"+1) fitting constraints and the fitting constraint term in the Lagrangian is then:


title: Logistic regression

text: where the "λ" are the appropriate Lagrange multipliers. There are "K" normalization constraints which may be written:


title: Logistic regression

text: where the "α" are the appropriate Lagrange multipliers. The Lagrangian is then the sum of the above three terms:


title: Logistic regression

text: The formula_279 are not all independent. We can add any constant -dimensional vector to each of the formula_279 without changing the value of the formula_275 probabilities so that there are only "N" rather than independent formula_279. In the section above, the formula_283 was subtracted from each formula_279 which set the exponential term involving formula_283 to unity, and the beta coefficients were given by formula_286.


title: Logistic regression

text: In machine learning applications where logistic regression is used for binary classification, the MLE minimises the Cross entropy loss function.


title: Logistic regression

text: Logistic regression is an important machine learning algorithm. The goal is to model the probability of a random variable formula_50 being 0 or 1 given experimental data.


title: Logistic regression

text: and since formula_291, we see that formula_292 is given by formula_293 We now calculate the likelihood function assuming that all the observations in the sample are independently Bernoulli distributed,


title: Logistic regression

text: Assuming the formula_296 pairs are drawn uniformly from the underlying distribution, then in the limit of large "N",


title: Logistic regression

text: where formula_298 is the conditional entropy and formula_299 is the Kullback–Leibler divergence. This leads to the intuition that by maximizing the log-likelihood of a model, you are minimizing the KL divergence of your model from the maximal entropy distribution. Intuitively searching for the model that makes the fewest assumptions in its parameters.


title: Logistic regression

text: Logistic regression can be seen as a special case of the generalized linear model and thus analogous to linear regression. The model of logistic regression, however, is based on quite different assumptions (about the relationship between the dependent and independent variables) from those of linear regression. In particular, the key differences between these two models can be seen in the following two features of logistic regression. First, the conditional distribution formula_300 is a Bernoulli distribution rather than a Gaussian distribution, because the dependent variable is binary. Second, the predicted values are probabilities and are therefore restricted to (0,1) through the logistic distribution function because logistic regression predicts the probability of particular outcomes rather than the outcomes themselves.


title: Logistic regression

text: A common alternative to the logistic model (logit model) is the probit model, as the related names suggest. From the perspective of generalized linear models, these differ in the choice of link function: the logistic model uses the logit function (inverse logistic function), while the probit model uses the probit function (inverse error function). Equivalently, in the latent variable interpretations of these two methods, the first assumes a standard logistic distribution of errors and the second a standard normal distribution of errors. Other sigmoid functions or error distributions can be used instead.


title: Logistic regression

text: Logistic regression is an alternative to Fisher's 1936 method, linear discriminant analysis. If the assumptions of linear discriminant analysis hold, the conditioning can be reversed to produce logistic regression. The converse is not true, however, because logistic regression does not require the multivariate normal assumption of discriminant analysis.


title: Logistic regression

text: The assumption of linear predictor effects can easily be relaxed using techniques such as spline functions.


title: Logistic regression

text: A detailed history of the logistic regression is given in . The logistic function was developed as a model of population growth and named "logistic" by Pierre François Verhulst in the 1830s and 1840s, under the guidance of Adolphe Quetelet; see for details. In his earliest paper (1838), Verhulst did not specify how he fit the curves to the data. In his more detailed paper (1845), Verhulst determined the three parameters of the model by making the curve pass through three observed points, which yielded poor predictions.


title: Logistic regression

text: The logistic function was independently developed in chemistry as a model of autocatalysis (Wilhelm Ostwald, 1883). An autocatalytic reaction is one in which one of the products is itself a catalyst for the same reaction, while the supply of one of the reactants is fixed. This naturally gives rise to the logistic equation for the same reason as population growth: the reaction is self-reinforcing but constrained.


title: Logistic regression

text: The logistic function was independently rediscovered as a model of population growth in 1920 by Raymond Pearl and Lowell Reed, published as , which led to its use in modern statistics. They were initially unaware of Verhulst's work and presumably learned about it from L. Gustave du Pasquier, but they gave him little credit and did not adopt his terminology. Verhulst's priority was acknowledged and the term "logistic" revived by Udny Yule in 1925 and has been followed since. Pearl and Reed first applied the model to the population of the United States, and also initially fitted the curve by making it pass through three points; as with Verhulst, this again yielded poor results.


title: Logistic regression

text: In the 1930s, the probit model was developed and systematized by Chester Ittner Bliss, who coined the term "probit" in , and by John Gaddum in , and the model fit by maximum likelihood estimation by Ronald A. Fisher in , as an addendum to Bliss's work. The probit model was principally used in bioassay, and had been preceded by earlier work dating to 1860; see . The probit model influenced the subsequent development of the logit model and these models competed with each other.


title: Logistic regression

text: The logistic model was likely first used as an alternative to the probit model in bioassay by Edwin Bidwell Wilson and his student Jane Worcester in . However, the development of the logistic model as a general alternative to the probit model was principally due to the work of Joseph Berkson over many decades, beginning in , where he coined "logit", by analogy with "probit", and continuing through and following years. The logit model was initially dismissed as inferior to the probit model, but "gradually achieved an equal footing with the logit", particularly between 1960 and 1970. By 1970, the logit model achieved parity with the probit model in use in statistics journals and thereafter surpassed it. This relative popularity was due to the adoption of the logit outside of bioassay, rather than displacing the probit within bioassay, and its informal use in practice; the logit's popularity is credited to the logit model's computational simplicity, mathematical properties, and generality, allowing its use in varied fields.


title: Logistic regression

text: The multinomial logit model was introduced independently in and , which greatly increased the scope of application and the popularity of the logit model. In 1973 Daniel McFadden linked the multinomial logit to the theory of discrete choice, specifically Luce's choice axiom, showing that the multinomial logit followed from the assumption of independence of irrelevant alternatives and interpreting odds of alternatives as relative preferences; this gave a theoretical foundation for the logistic regression.


title: Order of battle for the 2022 Russian invasion of Ukraine

text: Ukrainian military commentator Yuri Butusov presented the following initial February 2022 deployment of the Russian forces:


text: In April 2022, Army General Aleksandr Dvornikov was placed in overall charge of military operations in Ukraine, effectively ending the grouping system.


text: In June 2022 Colonel General Gennady Zhidko was appointed by Vladimir Putin to the top position, replacing Dvornikov.


text: In October 2022, Army General Sergey Surovikin, previously commander of Aerospace Forces and Army Group South, replaced Zhidko in overall command.


title: Mike Lindell

text: Michael James Lindell (born June 28, 1961), also known as the My Pillow Guy, is an American businessman, political activist, and conspiracy theorist. He is the founder and CEO of My Pillow, Inc., a pillow, bedding, and slipper manufacturing company.


title: Mike Lindell

text: Lindell is a prominent supporter of, and advisor to, former U.S. President Donald Trump. After Trump's defeat in the 2020 U.S. presidential election, Lindell played a significant role in supporting and financing Trump's attempts to overturn the election result; he spread disproven conspiracy theories about widespread electoral fraud in that election. He has also been an active promoter of the toxic plant extract oleandrin as an alternative medicine cure for COVID-19.


title: Mike Lindell

text: Lindell was born in 1961 in Mankato, Minnesota. He was raised in Chaska and Carver, Minnesota. Lindell's gambling addiction began to emerge in his teenage years. He attended the University of Minnesota after high school, but dropped out a few months into his studies. In his 20s, Lindell became addicted to and a frequent user of cocaine. This addiction became worse after he switched to crack cocaine in the 1990s. Lindell was also incurring gambling debts. The build up of his addictions between the 1980s and 1990s led to the foreclosure of his house and his wife filing for divorce. Lindell stated that he achieved sobriety through prayer in 2009.


title: Mike Lindell

text: In the 1980s, Lindell launched and operated a number of small businesses, including carpet cleaning, lunch wagons, and a few bars and restaurants in Carver County, Minnesota.


title: Mike Lindell

text: In 2004, Lindell invented the My Pillow pillow, which is filled with pieces of shredded foam that interlock. Lindell grew the business into a Minnesota manufacturing company.


title: Mike Lindell

text: In 2017, the Better Business Bureau (BBB) revoked accreditation of My Pillow, lowering its rating to an F based on a pattern of complaints by consumers. The BBB cited a buy one, get one free offer that became a continuous offer and therefore the normal price of the product, not a sale price or free offer. In a statement, Lindell said, "Naturally, I am terribly disappointed by the BBB's decision."


title: Mike Lindell

text: In 2020, Lindell named his son Darren as the company's chief operating officer, citing his own possible future political ambitions.


title: Mike Lindell

text: In 2021, some major retailers stopped carrying My Pillow products. Lindell has suggested that this is a result of his claims relating to the 2020 United States presidential election results, although outlets like Kohl's and Bed Bath & Beyond have stated this is due to market research and low customer demand.


title: Mike Lindell

text: Lindell operates two web sites under the "Frank" brand: Frank, focused on video streaming, and the social network FrankSocial.


title: Mike Lindell

text: In March 2021, Lindell detailed plans for "Vocl," an alt-tech social media platform that he had been developing for several months. He described the site as a cross between YouTube and Twitter that would be different from Gab and Parler. A dispute from a company that owned a web site called "Vocal" led Lindell to rename his site "Frank". Frank launched on the domain frankspeech.com on April 19, 2021, experiencing many technical issues, which Lindell ascribed to a "massive attack". Frank has no social networking features and primarily offers embedded video streams, including "Absolute Interference", a two-hour video promoting conspiracy theories about the 2020 presidential election.


title: Mike Lindell

text: Lindell has said he has spent millions of dollars developing Frank. According to invoices published by a "Salon" writer, obtained from a leaked video conference with Lindell's IT team, Lindell spent about $936,000 on hardware, labor, and services to launch Frank. Jared Holt, an extremism and far-right media researcher at the Atlantic Council's Digital Forensic Research Lab, opined that Lindell was "being had by the people around him... All the various products and ventures Lindell has going on, whether it's a pseudo-documentary film or a social media platform, are very expensive endeavors. Someone is taking Lindell's money from him to produce this stuff." Lindell told "Insider" in March 2022 that he was spending over $1 million per month on Frank.


title: Mike Lindell

text: FrankSocial, a social networking site hosted on a different domain, became available in April 2022. A reporter for "Insider" described FrankSocial as reminiscent of Facebook's design in 2012, with a basic news feed and no messaging function. Lindell had 308 followers there as of April 21, more than any other user.


title: Mike Lindell

text: During the early days of My Pillow, Lindell made donations to the Salvation Army, Union Gospel Mission, and other organizations.


title: Mike Lindell

text: Lindell founded the Lindell Foundation, a non-profit organization that assists former addicts in getting treatment and other services. The foundation broadened over time, including cancer victims and veterans in their outreach.


title: Mike Lindell

text: In 2019, Lindell launched the Lindell Recovery Network, which connects addicts with others who have gone through drug addiction and the recovery process, as well as faith-based treatment centers and other recovery organizations.


title: Mike Lindell

text: In August 2016, Lindell met with Republican presidential candidate Donald Trump, and became an avid supporter, calling Trump "the most amazing president this country has ever seen in history", following his 2016 election victory. In a speech at Liberty University in August 2019, Lindell said "When I met with Donald Trump, it felt like a divine appointment, and when I walked out of that office I decided I was going to go all in."


title: Mike Lindell

text: On October 19, 2016, Lindell attended the final presidential debate in Las Vegas. He spoke at a Trump campaign rally in Minneapolis on November 6, 2016, and attended the Official Donald Watch Party on November 8. He attended Trump's inauguration, and Trump gave him an inauguration lapel pin as a personal gift.


title: Mike Lindell

text: At a rally in Fargo, North Dakota, on June 27, 2018, Trump complimented Lindell for his "business acumen". Lindell spoke at a Trump rally on October 4, 2018, in Rochester, Minnesota. Lindell spoke at the 2019 Conservative Political Action Conference, in which he promoted Trump as "the greatest president in history" and "chosen by God".


title: Mike Lindell

text: In 2019, Lindell met with Trump and his staffer on issues of opioid addiction. He was there when Trump signed a bipartisan bill that addressed the growing opioid crisis and trying to prevent opioid abuse and death.


title: Mike Lindell

text: In a March 2020 appearance on Fox News, Lindell said that his company's bedding factories had been refocused on face mask production at the behest of the Trump administration. Later that month, Lindell appeared with Trump at a White House coronavirus press conference, at which Lindell praised Trump: "God gave us grace on November 8, 2016, to change the course we were on. God had been taken out of our schools and lives, a nation had turned its back on God. I encourage you to use this time at home to get back in the word. Read our Bible and spend time with our families."


title: Mike Lindell

text: Lindell has considered running for governor of Minnesota in 2022 against Democratic incumbent Tim Walz, reportedly at Trump's urging. He attended a Republican Governors Association meeting, at which he was encouraged to run. In May 2020, he became the campaign chair for Trump's reelection campaign in Minnesota. In July 2020, Lindell said he was "99% sure" about running for Minnesota governor; Lindell ultimately did not run.


title: Mike Lindell

text: In November 2020, Lindell was among those who paid for the bail of the Kenosha shooter, then-17-year-old Kyle Rittenhouse. Lindell responded to the accusations saying that he donated to "The Fight Back Foundation Inc. to help fund election fraud litigation, among other things" and that the story was "Fake News".


title: Mike Lindell

text: In April 2022, Lindell's mention of his donating as much as $800,000 to a legal defense fund for Tina Peters, a candidate for Colorado Secretary of State, raised questions as such a donation would violate Colorado state law; the state's ethics commission investigated the fund after a complaint about a lack of donor transparency.


title: Mike Lindell

text: In May 2022, Lindell rejoined Twitter, circumventing a Twitter ban on his former account. Hours after rejoining, however, Twitter terminated Lindell's newest account for ban evasion.


title: Mike Lindell

text: In November 2022, Lindell announced that he is running for Chair of the Republican National Committee.


title: Mike Lindell

text: In White House meetings with Trump and public appearances, Lindell has promoted the toxic plant extract oleandrin as a cure for COVID-19, saying, "This thing works – it's the miracle of all time." In a television appearance, Lindell made a misleading statement about the testing of the substance. Lindell has a financial stake in Phoenix Biotechnology, a company that makes oleandrin, and sits on its board. Lindell's unsubstantiated claims alarmed scientists, since there is no scientific evidence that oleandrin is a safe or effective coronavirus treatment, and there is evidence the plant is poisonous at low doses. After the efforts by Lindell and U.S. Secretary of Housing and Urban Development Ben Carson to promote oleandrin, Trump said his administration would "look at" the substance.


title: Mike Lindell

text: After Trump's defeat in the 2020 presidential election, Lindell played a significant role in supporting and financing Trump's attempts to overturn the election result.


title: Mike Lindell

text: Lindell helped sponsor a bus tour that sought to challenge the election results. The two-week tour ended in Washington, D.C., on December 14, 2020, and Lindell spoke at five stops along the way. Lindell said that "he did not help finance subsequent trips to promote the Jan. 6 rally," referring to the rally that was followed by the storming of the U.S. Capitol by a pro-Trump mob. Lindell attended this rally, but did not go to the Capitol. Following the Capitol storming, Lindell was among those who advanced the false conspiracy theory that people associated with Antifa were responsible for the attack, saying they had probably "dressed as Trump people". On January 15, 2021, Lindell carried a document into the West Wing of the White House; the document, as seen in an enhanced photo, appeared to refer to martial law and the Insurrection Act, as well as replacing Gina Haspel with Trump loyalist Kash Patel as the CIA director.


title: Mike Lindell

text: Lindell promoted a conspiracy theory, popular with Trump supporters, that falsely claimed that voting machine companies Smartmatic and Dominion conspired with foreign powers to rig voting machines to steal the election from Trump. In January and February 2021, Dominion warned Lindell that they planned to sue him, stating in their letter: "You have positioned yourself as a prominent leader of the ongoing misinformation campaign." (The two companies sent similar letters to others as well.) After Newsmax had broadcast a "clarification" of its false reporting on voting machine fraud in response to a demand letter from Dominion, followed by a defamation suit from a company executive, Lindell appeared on the network on February 2, 2021, to repeat the false claims, as network anchor Bob Sellers tried to stop Lindell before reading a disclaimer and walking off the set.


title: Mike Lindell

text: On January 25, 2021, Twitter permanently banned Lindell for perpetuating the unfounded claim that Trump won the 2020 election. A Twitter spokesperson explained that Lindell violated the company's civic integrity policy implemented in September 2020 to fight disinformation. After his personal account was banned, Lindell circumvented the ban by using the corporate My Pillow Twitter account to issue several tweets, including a call for Twitter CEO Jack Dorsey to "be found out and should be put in prison when all is revealed!" Following the tweets, Twitter permanently banned the My Pillow account for violating the platform's ban evasion policy.


title: Mike Lindell

text: Lindell purchased three hours of airtime on One America News Network, which had also been threatened with legal action, to broadcast "Absolute Proof", a documentary that makes false claims about the election, on February 5, 2021. OANN broadcast a lengthy disclaimer before the program saying the claims were Lindell's alone, but that the 2020 election results "remain disputed and questioned by millions of Americans". YouTube removed the documentary that day for violating its policy against election disinformation. At the 41st Golden Raspberry Awards, which gives awards to the films judged the worst, "Absolute Proof" won the Golden Raspberry Awards for Worst Picture and Worst Actor for Lindell. On April 20, 2021, Lindell published a second documentary titled "Absolute Interference", which also promoted debunked conspiracy theories about the election, and criticized the news media and communism.


title: Mike Lindell

text: Dominion sent cease-and-desist letters to Lindell in December 2020 and January 2021 and asked him to retract his false claims about the company. After Lindell refused (saying "sue me"), Dominion filed suit for defamation on February 22, 2021, naming both Lindell and My Pillow as defendants and seeking US$1.3 billion in damages. In its complaint, Dominion alleges that Lindell and My Pillow used "defamatory marketing campaign—with promo codes like 'FightforTrump,' '45,' 'Proof,' and 'QAnon'—[which have] increased My Pillow sales by 30–40% and [continued] duping people into redirecting their election-lie outrage into pillow purchases"; Dominion further stated its intent was to "stop Lindell and My Pillow from further profiting at Dominion's expense". On August 11, a federal court judge ruled that Dominion's defamation lawsuit against Lindell and My Pillow can proceed, noting that Dominion has "adequately alleged" that Lindell's accusations against the company were either knowingly false or made with "reckless disregard for the truth". Smartmatic sued Lindell for defamation in January 2022, seeking unspecified damages and legal fees, asserting his false claims had reduced the company's value from more than $3 billion before the 2020 election to less than $1 billion.


title: Mike Lindell

text: In March 2021, Lindell said on Steve Bannon's "War Room: Pandemic" podcast he would counter-sue Dominion in cases for himself and My Pillow, accusing Dominion of racketeering and election interference. On April 19, 2021, Lindell announced that My Pillow had filed a $1.6 billion lawsuit against Dominion, arguing that Dominion had defamed the company. Stephen Shackelford, legal counsel for Dominion, stated that My Pillow's lawsuit was "a meritless retaliatory lawsuit, filed by My Pillow to try to distract from the harm it caused to Dominion". Lindell also filed a civil suit against Dominion and Smartmatic in June 2021. The day after the suit was filed, a law partner with Barnes & Thornburg who drafted and submitted the suit left the firm, which said he did not have "firm authorization" to participate in the suit. In May 2022, the federal court dismissed all the claims by Lindell and MyPillow against Dominion, Smartmatic, and their lawyers and public relations firm; the court also ruled that Lindell and MyPillow would be liable to Smartmatic for attorney's fees, in an amount to be determined, for making claims that were frivolous under Rule 11.


title: Mike Lindell

text: Lindell and others predicted various inquiries into the election would result in Trump being reinstated as president in 2021. Lindell predicted that Trump would be reinstated on the morning of August 13, stating "it'll be the talk of the world". He held a three-day "Cyber Symposium" ending August 12, promising to present "irrefutable evidence" of election fraud, but none was produced; Lindell's own cybersecurity expert said that his purported evidence was a "pile of nothing" and found no proof of election fraud. After August 13 passed and president Joe Biden remained in office, Lindell moved his prediction for Trump's return to September 30.


title: Mike Lindell

text: In September 2021, Bonner County, Idaho, announced it would perform a recount of ballots cast in the 2020 presidential election, in response to an allegation by Lindell that all 44 Idaho counties had been digitally hacked. Lindell provided a detailed list of IP addresses he asserted had been compromised. County Clerk Mike Rosedale stated that all county voting machines were fully airgapped from the Internet, also noting that seven Idaho counties do not use voting machines. Lindell alleged that a specific formula had been applied by hackers to flip votes from Trump to Biden. Rosedale said Lindell had not contacted his office before presenting his allegations. Trump carried Bonner County with 67.2% of the vote and Idaho with 63.9% in the 2020 election. At the conclusion of the Idaho recount, accuracy was determined to be within 0.1% for the election results with no indications of digital hacking. Idaho plans to bill Lindell for the cost of the recount. The Bonner audit, and audits of two other counties that do not use voting machines, affirmed the accuracy of the ballot count. Chief Deputy Secretary of State Chad Houck said Lindell would be sent a bill for the audits.


title: Mike Lindell

text: In January 2022, Lindell claimed that he possessed "enough evidence to put everybody in prison for life, 300-some million people" for their part in the alleged 2020 election fraud, which, at the time, was about 91 percent of the U.S. population.


title: Mike Lindell

text: On September 13, 2022, the FBI served a search and seizure warrant for Lindell's phone. As recounted by Lindell, speaking on his web show, agents surrounded his car at a fast-food drive-thru in Minnesota. In addition to sharing the search warrant on his show, Lindell claims FBI agents asked questions regarding a Colorado county clerk charged with election interference earlier that month.


title: Mike Lindell

text: Lindell has been married twice. His first marriage ended in divorce after about 20 years, and he has children from this marriage. He married Dallas Yocum in June 2013 and filed for divorce in mid-July 2013 after she left him. Lindell stated that they had a prenuptial agreement.


title: Mike Lindell

text: Lindell is an evangelical Christian and received an honorary Doctor of Business from Liberty University in 2019, for his expertise as an entrepreneur. He self-published a book, "What Are the Odds? From Crack Addict to CEO," that year about his addiction recovery and his developing relationship with God.


title: Mike Lindell

text: In January 2021, the "Daily Mail" published a story stating that Lindell had a nine-month relationship with actress Jane Krakowski between late 2019 and the summer of 2020. Both Lindell and Krakowski denied the allegation. Lindell, represented by attorney Charles Harder, sued the "Daily Mail", alleging libel. A federal court dismissed the suit in December 2021, ruling that Lindell had failed to identify any statements "that a reasonable person would view as defamatory."


title: Mike Lindell

text: In March 2021, Lindell said on "The Domenick Nati Show" that he was no longer living in Minnesota and was not attending in-person events over safety concerns.


title: Darren Till

text: Darren Till (born 24 December 1992) is an English professional mixed martial artist and former Muay Thai fighter. He currently competes in the middleweight division of the Ultimate Fighting Championship (UFC). As of December 13, 2022, he is #15 in the UFC middleweight rankings.


title: Darren Till

text: Till was born in Liverpool, where he would frequently get into altercations. He began training in Muay Thai at the age of 12. He dropped out of high school at the age of 14 to focus on the sport, turning professional a year later and beginning to train in MMA with Team Kaobon at the age of 17. During his teenage years, he would live in different places such as his gym and relatives' houses after an incident he had with his mother which got him kicked out of the family home. In August 2012, he was stabbed twice in the back after confronting a large group of men at a party. The knife missed his main artery by 1 mm, with doctors calling him lucky to be alive.


title: Darren Till

text: Following the incident, Till's coach Colin Heron advised him to move to Brazil to train with a former Team Kaobon coach to avoid troubles in his hometown. He subsequently joined Astra Fight Team in Balneário Camboriú, despite being unable to speak Portuguese and with minimal knowledge of grappling. While he planned to spend six months in South America, he stayed for three and a half years and welcomed a daughter with his girlfriend in Brazil. In late 2016, he returned to England to be reunited with Heron at Team Kaobon. His daughter and then-girlfriend remained in Brazil; in 2018, he revealed he had not seen his daughter for over a year.


title: Darren Till

text: Till spent most of his early MMA career in Brazil under the guidance of Astra Fight Team. He went 3–0 as an amateur in England before making the move to South America, and turned professional in February 2013. Till had 11 fights in Brazil and one in Argentina prior to joining the UFC, with just two going to a decision. In 2013, he fought eight times as a middleweight before competing at welterweight in late 2014.


title: Darren Till

text: Till faced Brazilian Wendell de Oliveira Marques in his UFC debut on nine days' notice at on 30 May 2015. He knocked out Oliveira and secured his first UFC win.


title: Darren Till

text: Till next faced Nicolas Dalby on 24 October 2015 at . After a back and forth fight, the bout was scored a majority draw. Both participants were awarded the "Fight of the Night" bonus.


title: Darren Till

text: After a significant shoulder injury and various personal issues kept him out of action for an extended period of time, Till returned to face Jessin Ayari on 28 May 2017 at . He missed the non-title Welterweight limit of 171 pounds by 5 pounds, and had to give 20 percent of his fight earnings to his opponent. Till won the fight via unanimous decision.


title: Darren Till

text: Till faced Donald Cerrone on 21 October 2017 at . He won the fight via TKO in the first round. This win earned him the "Performance of the Night" bonus award.


title: Darren Till

text: Till faced former UFC Welterweight title challenger Stephen Thompson at on 27 May 2018. At the weigh-ins, Till weighed in at 174.5 pounds, 3.5 pounds over the Welterweight non-title fight limit of 171. After negotiating with Thompson's team, the bout proceeded at a catchweight with the stipulation that Till could not weigh more than 188 pounds on the day of the fight. Till also forfeited 30 percent of his purse to Thompson. Till won the fight via a controversial unanimous decision. 22 of 25 media outlets scored the bout in favor of Thompson.


title: Darren Till

text: Till was given the opportunity to face Tyron Woodley on 8 September 2018 at UFC 228, challenging him for the UFC Welterweight Championship.


title: Darren Till

text: He lost the fight via D'Arce choke submission in the second round. This was Till's first loss in his MMA career.


title: Darren Till

text: Despite talks of a potential move up to Middleweight, Till stayed at Welterweight and faced Jorge Masvidal on 16 March 2019 at UFC Fight Night 147. Till lost the fight by knockout in the second round. Both participants were awarded a "Fight of the Night" bonus award.


title: Darren Till

text: On 26 August, it was announced Till was returning to the middleweight division to face Kelvin Gastelum at UFC 244 on 2 November. He won the fight via split decision.


title: Darren Till

text: Till faced Robert Whittaker on 26 July 2020 at . He lost the fight via unanimous decision, though many admired Till's performance against a former title holder in Whittaker.


title: Darren Till

text: Till was scheduled to face Jack Hermansson on 5 December 2020 at UFC on ESPN 19. However, it was announced on 6 November that Till was forced to withdraw from the bout due to an injury. He was replaced by Kevin Holland, who was later replaced by Marvin Vettori. Till was forced to pull out from the match due to an undisclosed injury.


title: Darren Till

text: Till was scheduled to face Marvin Vettori on 10 April 2021, at UFC on ABC 2 On the week before the fight, Till was forced to pull out from the match due to a broken collarbone, and he was replaced by Kevin Holland.


title: Darren Till

text: Till faced Derek Brunson on 4 September 2021, at . He lost the fight via rear-naked choke submission in round three.


title: Darren Till

text: Till was scheduled to face Jack Hermansson on 23 July 2022, at UFC Fight Night 208. He was replaced by Chris Curtis after being forced to withdraw from the bout due to an undisclosed injury.


title: Darren Till

text: Till faced Dricus du Plessis on 10 December 2022, at UFC 282. He lost the fight via submission in round three. This fight earned him the "Fight of the Night" award.


title: Darren Till

text: On 18 April 2019, Till was arrested in Tenerife after trashing a hotel room when he was asked to leave, and for also stealing a taxi while the driver was loading his luggage into the trunk. Till and four other involved individuals were sentenced for a crime of damage and unauthorised use of a vehicle, and were fined approximately £10,000.


title: Darren Till

text: Till was penalized by Swedish police for driving under the influence of alcohol with over three times the legal limit in his system on 31 July 2022.


title: Leslie Odom Jr.

text: Leslie Lloyd Odom Jr. (; born August 6, 1981) is an American actor and singer. He made his acting debut on Broadway in 1998 and first gained recognition for his portrayal of Aaron Burr in the musical "Hamilton", which earned him a 2016 Tony Award for Best Actor in a Leading Role in a Musical and a Grammy Award for Best Musical Theater Album in the same year.


title: Leslie Odom Jr.

text: Odom is also known for his roles in the television series "Smash" (2012–2013) and "Person of Interest" (2013–2014), as well as the films "Murder on the Orient Express" (2017), "Harriet" (2019), and "Hamilton" (2020). Odom currently voices the character of Owen Tillerman in the Apple TV+ animated musical-comedy series "Central Park," for which he was nominated for an Emmy Award in 2020.


title: Leslie Odom Jr.

text: For his performance as singer Sam Cooke in "One Night in Miami..." (2020), Odom earned Academy Award, BAFTA Award, Screen Actors Guild Award, Critics' Choice Movie Award, and Golden Globe Awards nominations for Best Supporting Actor. He was also nominated for an Oscar and Golden Globe for writing the film's original song "Speak Now," for which he won Best Song at the 2021 Critics' Choice Awards. Later that year, he received a nomination for a Primetime Emmy Award for Outstanding Actor in a Leading Role in a Limited Series or Movie for his performance in the 2020 Disney+ live stage recording of "Hamilton". Also in 2021, he received critical praise for his role as Harold McBrayer in "The Sopranos" prequel film "The Many Saints of Newark".


title: Leslie Odom Jr.

text: Odom was born in Queens, New York City. His father, Leslie Lloyd Odom Sr., worked in sales. One of his maternal great-grandfathers was from South Africa, and a maternal great-great-grandfather was from Bridgetown, Barbados. Odom's family moved to the East Oak Lane section of Philadelphia, where he grew up. He attended Julia R. Masterman School for middle school and Philadelphia High School for Creative and Performing Arts for high school. Odom and his family attended Canaan Baptist Church in the Germantown section of Philadelphia, where he sang solos in the church choir.


title: Leslie Odom Jr.

text: He earned a degree with honors at Carnegie Mellon University in Pittsburgh, Pennsylvania, and then moved to Los Angeles in the summer of 2003.


title: Leslie Odom Jr.

text: Odom attended the New Freedom Theatre where he studied theater and performed in musicals. He then went on to study musical theater in college. At the age of 17 he made his Broadway debut as Paul in "Rent". In 2001, he appeared in the ensemble of the one-night Broadway concert version of "Dreamgirls".


title: Leslie Odom Jr.

text: He spent much of his time doing theatre in Los Angeles, including in 2010 with "Leap of Faith", which moved to Broadway in 2012, with him starring as Isaiah Sturdevant. When the show closed shortly afterward, he relocated to New York City.


title: Leslie Odom Jr.

text: After starring Off-Broadway in "Venice" and in "Witness Uganda" (later retitled "Invisible Thread") at A.R.T. and in workshops, Odom worked with Lin-Manuel Miranda in the Encores! Off-Center production of "Tick, Tick... Boom!", playing Michael. Odom also played Nat King Cole in the 2015 one-night Actors Fund of America benefit concert of "Bombshell".


title: Leslie Odom Jr.

text: Odom was nominated for a 2015 Drama Desk Award for Outstanding Featured Actor in a Musical for the role of Aaron Burr in the off-Broadway production of "Hamilton" at The Public Theater. He continued in the same role at the Richard Rodgers Theatre after the show transferred to Broadway later that year. He also won a 2016 Grammy Award for the cast album, and won the Tony Award for Best Actor in a Musical.


title: Leslie Odom Jr.

text: He has made guest appearances on several shows including "Gilmore Girls", "Grey's Anatomy", "Supernatural", and "The Good Wife". Odom had a recurring role on "", appearing as Joseph Kayle in nine episodes between 2003 and 2006.


title: Leslie Odom Jr.

text: He followed with a recurring role on "Vanished", appearing in 10 episodes as Maliko Christo. From 2006 to 2007, he held a recurring role as Freddy on "Big Day", appearing on nine episodes prior to its cancellation. He also had a supporting role in the TV movie "Poe" in 2011.


title: Leslie Odom Jr.

text: In the musical television series "Smash", Odom had a recurring role as Sam Strickland in 2012, and was promoted to a starring role for the show's final season in 2013.


title: Leslie Odom Jr.

text: In late 2013, Odom was offered the leading role of Lucas Newsome on "State of Affairs". He accepted the role, but ended up backing out to take the part of Aaron Burr in the workshops of the musical "Hamilton".


title: Leslie Odom Jr.

text: In 2013 and 2014, he had a recurring role on "Person of Interest" as Peter Collier, appearing in eight episodes. He also played Reverend Curtis Scott in seven episodes of "" between 2013 and 2015.


title: Leslie Odom Jr.

text: Beginning on May 19, 2016, Odom hosted the Broadway.com web series "Aaron Burr, Sir: Backstage at Hamilton with Leslie Odom Jr.", with a new episode being uploaded every Thursday for eight weeks. Odom was also set to play Jimmy Jam in a 2017 miniseries about New Edition, but ended up leaving the project.


title: Leslie Odom Jr.

text: In 2020, Odom starred and executive produced the four-part, television miniseries "Love in the Time of Corona" opposite his wife Nicolette Robinson. "Love in the Time of Corona" follows four interwoven stories about the hopeful search for love and connection during the quarantine as a result of the COVID-19 pandemic. Production began virtually on June 29, 2020, in Los Angeles using remote technologies. The limited series premiered on Freeform on August 22, 2020.


title: Leslie Odom Jr.

text: Odom currently voices the character of Owen Tillerman in the Apple TV+ animated musical-comedy series "Central Park," for which he was nominated for Outstanding Character Voice-Over Performance at the 2020 Primetime Emmy Awards. He will also guest star on the Disney+ "The Proud Family" continuation "" in its second season as Kwame.


title: Leslie Odom Jr.

text: Odom has appeared in short films, and had a supporting role as Walter Hall in the 2012 war film "Red Tails".


title: Leslie Odom Jr.

text: He played Dr. Arbuthnot in Kenneth Branagh's 2017 adaptation of "Murder on the Orient Express". He also starred as abolitionist William Still in the 2019 film "Harriet", about abolitionist Harriet Tubman. In 2019, Odom starred in "Only", a post-apocalyptic romance alongside Freida Pinto.


title: Leslie Odom Jr.

text: In 2021, Odom starred as Ebo in musical film "Music", which was co-written and directed by Australian singer-songwriter Sia. It was released in February 2021.


title: Leslie Odom Jr.

text: In 2020, Odom was cast as soul singer Sam Cooke in the Regina King–directed film adaptation of "One Night in Miami...", which was released in theaters and Amazon Prime on January 15, 2021. Odom received rave reviews from critics for his portrayal of Cooke and has since earned Academy Award, Golden Globe, and SAG Award nominations including for the Best Supporting Actor and an additional nomination for Outstanding Performance by a Cast in a Motion Picture. With Sam Ashworth as co-writer, Odom co-wrote and performed "Speak Now", which also received a nomination for Best Original Song at the 78th Golden Globe Awards to be held on February 28, 2021. "Speak Now" is played during the end credits of "One Night in Miami".


title: Leslie Odom Jr.

text: In May 2021, Odom was cast in the "", the sequel to Rian Johnson's "Knives Out" starring Daniel Craig as Benoit Blanc. In July 2021, Odom was cast for the upcoming, as yet untitled sequel film to William Friedkin's "The Exorcist", with Ellen Burstyn reprising her role as Chris MacNeil.


title: Leslie Odom Jr.

text: In 2014, Odom self-released his debut album as a jazz singer, "Leslie Odom Jr." on CD and SoundCloud, and promoted the album with several concerts at The Public Theater. In February 2016, he released a version of Selena Gomez's "Good For You" featuring his "Hamilton" castmate Daveed Diggs.


title: Leslie Odom Jr.

text: Odom was signed to a four-album deal with S-Curve Records in 2016. He and producer Steve Greenberg narrowed down 200 potential tunes to ten tracks, and Odom recorded an updated and improved version of "Leslie Odom Jr." during days off and afternoons before Broadway performances, in order to release the album before Odom left "Hamilton". Released in June 2016, the album charted at No. 1 on Billboard Jazz and No. 147 on Billboard 200.


title: Leslie Odom Jr.

text: On November 11, 2016, Odom released his second album, "Simply Christmas", which featured jazz interpretations of Christmas standards. A deluxe edition of the album was released in October 2017, with four additional songs. "Simply Christmas" hit #1 on iTunes and the Billboard Jazz charts, reaching No. 4 on the Billboard Holiday chart and No. 31 on the Billboard Top 200 chart.


title: Leslie Odom Jr.

text: Odom released his third album and first of original material, "Mr," in November 2019, and in October 2020, he debuted a new version of the album's standout song "Cold", featuring Sia.


title: Leslie Odom Jr.

text: His critically acclaimed second holiday album, "The Christmas Album," was released in November 2020. "The Christmas Album" is Odom's fourth studio album.


title: Leslie Odom Jr.

text: Odom tours to promote his albums, performing concerts backed by a jazz quintet that includes a drummer, percussionist, bassist, guitarist and a pianist who is also Odom's musical director.


title: Leslie Odom Jr.

text: In December 2017, Odom returned to the New York City stage in a solo concert at Jazz at Lincoln Center. The cabaret-style performance was crafted around signature songs and music that shaped this artist's journey, all performed with a world-class band in front of a live audience. The show was filmed for broadcast as an hour-long PBS special as part of the 17-time Emmy Award-winning series, "Live from Lincoln Center", and premiered in April 2018.


title: Leslie Odom Jr.

text: In June 2017, it was announced that Odom had a book deal for "Failing Up: How to Take Risks, Aim Higher, and Never Stop Learning", which he was writing with the intention to inspire younger readers. The book, by an imprint of Macmillan in March 2018, "outlines the setbacks and rejections that preceded his success." Odom modeled his writing on the style of a commencement speech, exploring what he had learned throughout his life and the importance of pursuing passions.


title: Leslie Odom Jr.

text: Odom has been married to actress Nicolette Kloe Robinson since December 1, 2012. Their daughter, Lucille Ruby, was born on April 23, 2017. Their son, Able Phineas, was born on March 25, 2021.


title: Leslie Odom Jr.

text: The couple met in 2008 while Robinson was a student at UCLA and auditioned for "Once on This Island" in Los Angeles. When she replaced a cast member who left unexpectedly, assistant director Odom became responsible for bringing her quickly up to speed, and a romantic relationship ensued. Robinson has since appeared in the off-Broadway musical "Invisible Thread" (2015) and the television series "The Affair", and played the lead role in the Broadway musical "Waitress" from September 4 through December 9, 2018.


title: Leslie Odom Jr.

text: In 2019, Odom was conferred an honorary Doctor of Fine Arts by his alma mater, Carnegie Mellon University, at its 122nd Commencement, where Odom was also the keynote speaker.


title: Jaleel White

text: Jaleel Ahmad White (born November 27, 1976) is an American actor. He is best known for his role as Steve Urkel on the sitcom "Family Matters".