R: How to or should I drop an insignificant orthogonal polynomial basis in a linear model?
<p>I have soil moisture data with x-, y- and z-coordinates like this:</p>

<pre><code>gue &lt;- structure(list(x = c(311939.1507, 311935.4607, 311924.7316, 311959.553, 
                     311973.5368, 311953.3743, 311957.9409, 311948.3151, 311946.7169, 
                     311997.0803, 312017.5236, 312006.0245, 312001.5179, 311992.7044, 
                     311977.3076, 311960.4159, 311970.6047, 311957.2564, 311866.4246, 
                     311870.8714, 311861.4461, 311928.7096, 311929.6291, 311929.4233, 
                     311891.2915, 311890.3429, 311900.8905, 311864.4995, 311870.8143, 
                     311866.9257, 312002.571, 312017.816, 312004.5024, 311947.1186, 
                     311943.0152, 311952.2695, 311920.6095, 311929.8371, 311918.6095, 
                     312011.9019, 311999.5755, 312011.1461, 311913.7251, 311925.3459, 
                     311944.4701, 311910.2079, 311908.7618, 311896.0776, 311864.4814, 
                     311856.9027, 311857.5747, 311967.3779, 311962.2024, 311956.8318, 
                     311977.5254, 311971.1776, 311982.537, 311993.4709, 312004.6407, 
                     312015.6118, 311990.8601, 311994.686, 311988.3037, 311990.518, 
                     311986.3918, 311998.8876, 311923.9157, 311903.4563, 311915.714, 
                     311856.9087, 311858.9812, 311874.5867, 311963.9099, 311938.4542, 
                     311945.9505, 311804.3039, 311797.2571, 311791.6967, 311921.3965, 
                     311928.9353, 311920.0597, 311833.5109, 311829.8683, 311847.6261, 
                     311889.1243, 311902.4909, 311901.245, 311981.1118, 312005.7098, 
                     311976.5858, 311819.8901, 311816.4143, 311819.4172, 311870.418, 
                     311873.2656, 311888.3401, 311910.8377, 311897.6697, 311902.4571, 
                     311846.8196, 311833.6235, 311846.2942, 311931.3916, 311930.1891, 
                     311947.659, 311792.2642, 311793.2539, 311794.1931, 311795.1288, 
                     311796.0806, 311797.0142, 311797.95, 311798.8822, 311799.8229, 
                     311800.7774, 311801.7094, 311802.6395, 311803.583, 311804.5185, 
                     311805.4558, 311806.391, 311807.3346, 311808.2757, 311809.2187, 
                     311810.1549, 311811.1014, 311812.0366, 311812.9667, 311813.9107, 
                     311814.8373, 311815.7777, 311816.7365, 311817.6522, 311818.6091, 
                     311819.5335, 311820.4961, 311821.4337, 311822.3855, 311823.3195, 
                     311824.2713, 311825.214, 311826.1705, 311827.1188, 311828.0501, 
                     311828.9893, 311829.9324, 311830.8706, 311831.8181, 311832.7667, 
                     311833.705, 311834.6546, 311835.609, 311836.5527, 311837.5157, 
                     311838.4495, 311839.3926, 311840.3423, 311841.2799, 311842.2288, 
                     311843.1691, 311844.118, 311845.0746, 311846.019, 311846.9709, 
                     311847.9201, 311848.859, 311849.8105, 311850.7503, 311851.6889, 
                     311852.6355, 311853.6045, 311854.5296, 311855.4717, 311856.4171, 
                     311857.3759, 311858.3151, 311859.2604, 311860.2178, 311861.1636, 
                     311862.1071, 311863.0347, 311863.9857, 311864.9316, 311865.8722, 
                     311866.8158, 311867.7702, 311868.7155, 311869.649, 311870.6018, 
                     311871.5449, 311872.4871, 311873.4352, 311874.385, 311875.3042, 
                     311876.2617, 311877.2068, 311878.1429, 311879.0956, 311880.0401, 
                     311880.9822, 311881.929, 311882.8651, 311883.8017, 311884.7429, 
                     311885.6949, 311886.6349, 311887.7207, 311888.6653, 311889.6041, 
                     311890.5358, 311891.4838, 311892.4292, 311893.3736, 311894.326, 
                     311895.2703, 311896.2182, 311897.1635, 311898.1032, 311899.0496, 
                     311899.9967, 311900.9456, 311901.8889, 311902.8162, 311903.7566, 
                     311904.6996, 311905.6627, 311906.5899, 311907.5448, 311908.4856, 
                     311909.4399, 311910.3649, 311911.3188, 311912.2629, 311913.2022, 
                     311914.1527, 311915.1025, 311916.0425, 311916.985, 311917.9254, 
                     311918.8661, 311919.8174, 311920.7668, 311921.7026, 311922.6517, 
                     311923.5949, 311924.5252, 311925.4599, 311926.422, 311927.3646, 
                     311928.3, 311929.2432, 311930.1796, 311931.1358, 311932.0726, 
                     311933.0069, 311933.9585, 311934.845, 311935.7788, 311936.7193, 
                     311937.6441, 311938.572, 311939.5094, 311940.4666, 311941.4067, 
                     311942.3489, 311943.2712, 311944.2195, 311945.1536, 311946.0927, 
                     311947.0413, 311947.9761, 311948.9082, 311949.8557, 311950.8201, 
                     311951.7616, 311952.7148, 311953.7894, 311954.7289, 311955.6646, 
                     311956.6081, 311957.5588, 311958.4896, 311959.4297, 311960.3761, 
                     311961.3191, 311962.2653, 311963.195, 311964.1501, 311965.0856, 
                     311966.0254, 311966.9739, 311967.9305, 311968.8592, 311971.7861, 
                     311970.758, 311969.8205), y = c(5846548.408, 5846546.489, 5846538.014, 
                                                     5846525.283, 5846510.302, 5846503.516, 5846529.769, 5846523.06, 
                                                     5846522.742, 5846512.263, 5846525.347, 5846522.042, 5846537.487, 
                                                     5846545.587, 5846532.112, 5846425.917, 5846406.543, 5846434.03, 
                                                     5846500.989, 5846498.286, 5846487.134, 5846488.045, 5846483.29, 
                                                     5846468.713, 5846534.269, 5846533.527, 5846504.056, 5846453.395, 
                                                     5846438.43, 5846442.608, 5846406.8, 5846434.58, 5846419.229, 
                                                     5846441.045, 5846436.903, 5846447.917, 5846460.757, 5846457.428, 
                                                     5846451.067, 5846445.596, 5846474.031, 5846457.239, 5846532.694, 
                                                     5846553.938, 5846565.323, 5846446.926, 5846432.549, 5846467.236, 
                                                     5846473.963, 5846464.78, 5846498.142, 5846458.168, 5846474.018, 
                                                     5846489.801, 5846559.513, 5846589.975, 5846555.723, 5846553.847, 
                                                     5846560.066, 5846560.792, 5846455.642, 5846546.374, 5846465.999, 
                                                     5846432.091, 5846422.061, 5846442.871, 5846485.956, 5846472.811, 
                                                     5846506.756, 5846416.327, 5846419.623, 5846413.124, 5846587.334, 
                                                     5846600.116, 5846589.515, 5846463.69, 5846456.712, 5846459.683, 
                                                     5846600.118, 5846574.99, 5846597.804, 5846419.496, 5846437.615, 
                                                     5846436.902, 5846567.872, 5846572.857, 5846556.904, 5846388.146, 
                                                     5846393.088, 5846390.13, 5846481.09, 5846496.127, 5846493.586, 
                                                     5846545.396, 5846532.126, 5846538.334, 5846388.343, 5846416.117, 
                                                     5846392.223, 5846513.526, 5846486.644, 5846512.917, 5846395.509, 
                                                     5846386.421, 5846383.873, 5846459.062, 5846459.36, 5846459.682, 
                                                     5846460.026, 5846460.377, 5846460.703, 5846461.047, 5846461.378, 
                                                     5846461.73, 5846462.071, 5846462.418, 5846462.765, 5846463.115, 
                                                     5846463.466, 5846463.815, 5846464.128, 5846464.505, 5846464.843, 
                                                     5846465.189, 5846465.52, 5846465.869, 5846466.217, 5846466.557, 
                                                     5846466.893, 5846467.237, 5846467.586, 5846467.903, 5846468.274, 
                                                     5846468.601, 5846468.943, 5846469.258, 5846469.592, 5846469.909, 
                                                     5846470.247, 5846470.565, 5846470.891, 5846471.24, 5846471.536, 
                                                     5846471.885, 5846472.224, 5846472.553, 5846472.884, 5846473.225, 
                                                     5846473.532, 5846473.89, 5846474.179, 5846474.502, 5846474.827, 
                                                     5846475.146, 5846475.448, 5846475.768, 5846476.102, 5846476.428, 
                                                     5846476.746, 5846477.069, 5846477.37, 5846477.685, 5846478.009, 
                                                     5846478.335, 5846478.656, 5846478.958, 5846479.299, 5846479.608, 
                                                     5846479.926, 5846480.267, 5846480.603, 5846480.908, 5846481.246, 
                                                     5846481.56, 5846481.877, 5846482.19, 5846482.503, 5846482.825, 
                                                     5846483.144, 5846483.468, 5846483.811, 5846484.13, 5846484.458, 
                                                     5846484.8, 5846485.125, 5846485.456, 5846485.778, 5846486.112, 
                                                     5846486.421, 5846486.75, 5846487.08, 5846487.401, 5846487.744, 
                                                     5846488.067, 5846488.39, 5846488.728, 5846489.067, 5846489.383, 
                                                     5846489.716, 5846490.054, 5846490.38, 5846490.719, 5846491.044, 
                                                     5846491.357, 5846491.694, 5846492.005, 5846492.402, 5846492.726, 
                                                     5846493.045, 5846493.389, 5846493.708, 5846494.049, 5846494.363, 
                                                     5846494.686, 5846494.982, 5846495.3, 5846495.64, 5846495.957, 
                                                     5846496.263, 5846496.584, 5846496.911, 5846497.241, 5846497.591, 
                                                     5846497.914, 5846498.226, 5846498.553, 5846498.893, 5846499.221, 
                                                     5846499.538, 5846499.869, 5846500.19, 5846500.508, 5846500.82, 
                                                     5846501.151, 5846501.492, 5846501.827, 5846502.147, 5846502.471, 
                                                     5846502.803, 5846503.129, 5846503.46, 5846503.783, 5846504.11, 
                                                     5846504.448, 5846504.76, 5846505.118, 5846505.445, 5846505.79, 
                                                     5846506.106, 5846506.465, 5846506.795, 5846507.118, 5846507.448, 
                                                     5846507.758, 5846508.081, 5846508.396, 5846508.645, 5846508.99, 
                                                     5846509.34, 5846509.685, 5846510.031, 5846510.363, 5846510.693, 
                                                     5846511.031, 5846511.362, 5846511.694, 5846512.024, 5846512.354, 
                                                     5846512.701, 5846513.034, 5846513.353, 5846513.683, 5846513.998, 
                                                     5846514.32, 5846514.636, 5846514.956, 5846515.326, 5846515.65, 
                                                     5846515.968, 5846516.301, 5846516.634, 5846516.971, 5846517.318, 
                                                     5846517.64, 5846517.952, 5846518.308, 5846518.626, 5846518.937, 
                                                     5846519.27, 5846519.597, 5846519.921, 5846520.245, 5846520.581, 
                                                     5846521.498, 5846521.209, 5846520.893), z = c(26.485, 26.411, 
                                                                                                   26.339, 27.248, 27.208, 26.799, 27.199, 27.023, 26.973, 26.908, 
                                                                                                   26.275, 26.474, 26.316, 26.226, 27.184, 25.903, 25.765, 25.931, 
                                                                                                   26.057, 26.181, 26.102, 26.436, 26.457, 26.396, 25.585, 25.572, 
                                                                                                   26.448, 25.637, 25.603, 25.634, 25.847, 26.185, 25.899, 26.016, 
                                                                                                   25.873, 26.299, 26.358, 26.344, 26.088, 26.264, 26.3, 26.306, 
                                                                                                   26.311, 25.857, 26.004, 25.824, 25.798, 26.326, 26.03, 25.625, 
                                                                                                   25.78, 26.368, 26.225, 26.582, 26.398, 25.343, 26.253, 25.908, 
                                                                                                   25.323, 25.381, 26.3, 26.179, 26.284, 26.024, 25.896, 26.251, 
                                                                                                   26.447, 26.385, 26.419, 25.188, 25.176, 25.169, 25.348, 25.188, 
                                                                                                   25.291, 25.285, 25.266, 25.262, 25.333, 25.308, 25.314, 25.145, 
                                                                                                   25.172, 25.22, 25.235, 25.204, 25.286, 25.155, 25.397, 25.202, 
                                                                                                   25.373, 25.327, 25.341, 25.172, 25.253, 25.318, 25.023, 25.24, 
                                                                                                   25.132, 25.264, 25.38, 25.221, 25.119, 25.179, 25.083, 25.258, 
                                                                                                   25.254, 25.235, 25.252, 25.266, 25.256, 25.264, 25.26, 25.262, 
                                                                                                   25.265, 25.265, 25.285, 25.28, 25.257, 25.254, 25.258, 25.287, 
                                                                                                   25.294, 25.282, 25.27, 25.268, 25.309, 25.303, 25.3, 25.312, 
                                                                                                   25.305, 25.3, 25.314, 25.319, 25.328, 25.304, 25.325, 25.308, 
                                                                                                   25.332, 25.333, 25.333, 25.346, 25.344, 25.339, 25.355, 25.362, 
                                                                                                   25.36, 25.391, 25.418, 25.434, 25.436, 25.447, 25.486, 25.5, 
                                                                                                   25.526, 25.552, 25.551, 25.564, 25.589, 25.606, 25.641, 25.672, 
                                                                                                   25.689, 25.709, 25.736, 25.758, 25.782, 25.836, 25.844, 25.866, 
                                                                                                   25.88, 25.935, 25.984, 26.037, 26.066, 26.071, 26.094, 26.106, 
                                                                                                   26.106, 26.118, 26.1, 26.146, 26.135, 26.156, 26.169, 26.162, 
                                                                                                   26.173, 26.198, 26.196, 26.228, 26.258, 26.276, 26.283, 26.277, 
                                                                                                   26.236, 26.277, 26.251, 26.264, 26.26, 26.261, 26.249, 26.307, 
                                                                                                   26.289, 26.243, 26.206, 26.231, 26.224, 26.238, 26.244, 26.245, 
                                                                                                   26.254, 26.2, 26.229, 26.24, 26.248, 26.223, 26.29, 26.344, 26.371, 
                                                                                                   26.364, 26.311, 26.343, 26.342, 26.334, 26.317, 26.342, 26.315, 
                                                                                                   26.312, 26.322, 26.325, 26.324, 26.32, 26.308, 26.329, 26.31, 
                                                                                                   26.32, 26.327, 26.34, 26.371, 26.442, 26.442, 26.483, 26.504, 
                                                                                                   26.526, 26.562, 26.562, 26.538, 26.534, 26.533, 26.541, 26.584, 
                                                                                                   26.642, 26.65, 26.691, 26.719, 26.755, 26.786, 26.794, 26.849, 
                                                                                                   26.867, 26.919, 26.93, 26.945, 26.947, 26.959, 26.984, 26.992, 
                                                                                                   27.006, 27.035, 27.021, 27.052, 27.094, 27.104, 27.119, 27.16, 
                                                                                                   27.182, 27.223, 27.236, 27.267, 27.304, 27.331, 27.348, 27.341, 
                                                                                                   27.379, 27.355, 27.378, 27.357, 27.373, 27.319, 27.299, 27.278, 
                                                                                                   27.28, 27.295, 27.288, 27.286, 27.279), soil_m_sat = c(24.1, 
                                                                                                                                                          24.2, 26.9, 13.9, 20.6, 34.1, 16.2, 16.7, 16, 22.1, 23.9, 27.2, 
                                                                                                                                                          26.8, 34.4, 26.3, 54.1, 51, 44.9, 46.4, 45.9, 54.7, 39.1, 38.7, 
                                                                                                                                                          40.7, 56.5, 56.3, 40.6, 60.9, 56.8, 56.3, 40.7, 40.4, 44.1, 44.9, 
                                                                                                                                                          46.2, 45.3, 46.1, 43.7, 44.9, 45.4, 33.1, 45.8, 27.6, 47.8, 37.3, 
                                                                                                                                                          58.9, 51.4, 42.1, 46, 66.6, 51.1, 31.6, 48.7, 32.9, 28.1, 84, 
                                                                                                                                                          37.7, 38.2, 80.4, 73.3, 35.6, 44.2, 39.7, 50.2, 49.9, 37.8, 37, 
                                                                                                                                                          41.7, 27.3, 100, 100, 100, 80.9, 100, 88.4, 89.6, 93.8, 95.3, 
                                                                                                                                                          91.9, 93.9, 96.1, 91.4, 100, 94.4, 100, 100, 80, 94.1, 84.4, 
                                                                                                                                                          91.1, 80, 78.9, 85.9, 100, 97.5, 87.2, 88.6, 83.3, 90.7, 100, 
                                                                                                                                                          82.2, 100, 96.3, 93.3, 99.6, 92.1, 92.8, 90.9, 92.3, 91.2, 94.5, 
                                                                                                                                                          91.8, 89.4, 87, 86, 88, 83.7, 88.8, 92.9, 89.3, 83.3, 83.5, 84.5, 
                                                                                                                                                          85.8, 87.4, 86.5, 82, 78.1, 85.8, 85.6, 88.7, 87.7, 84.9, 82, 
                                                                                                                                                          87.9, 85.5, 86, 82, 83, 88.5, 81.2, 81.6, 76.5, 77.6, 84.5, 81.5, 
                                                                                                                                                          82, 82.4, 68, 67.7, 62.1, 68.9, 61.7, 68.5, 68.6, 65.3, 59.5, 
                                                                                                                                                          60.8, 67.3, 66.2, 59.9, 50.9, 46.9, 44.6, 47.9, 53, 52.1, 48.3, 
                                                                                                                                                          41.3, 53.8, 51, 47, 53.7, 49.5, 51.1, 44.4, 35.1, 42.2, 41.5, 
                                                                                                                                                          40, 48.2, 46.7, 48.6, 51.7, 51.2, 52.3, 53.4, 48.9, 50.7, 48.5, 
                                                                                                                                                          46.5, 39.4, 38, 49.2, 43.6, 47.1, 40.4, 44.7, 45.7, 38.1, 41.9, 
                                                                                                                                                          39.3, 40.2, 43.8, 47.3, 50.1, 41.2, 39.8, 46, 40.8, 40, 37.8, 
                                                                                                                                                          42.6, 46, 43.8, 45.4, 42.2, 46.5, 40.4, 39.9, 53, 44.7, 35.8, 
                                                                                                                                                          42.9, 43.9, 43.2, 40.6, 40.8, 32.2, 32.6, 33.5, 36.7, 34.6, 34.7, 
                                                                                                                                                          50.9, 35.6, 34.2, 28.1, 42, 32, 42.3, 30, 29.6, 31, 29.8, 26, 
                                                                                                                                                          37.8, 40, 37, 30.2, 28.2, 26.2, 27.4, 22.1, 28.4, 23.2, 24.8, 
                                                                                                                                                          26.5, 23.9, 21.1, 27.2, 20.8, 12.5, 14, 17.9, 19.7, 19.4, 26, 
                                                                                                                                                          16.7, 18.2, 23.9, 19, 25.9, 24.4, 22.1, 19.2, 18.4, 24.7, 17.3, 
                                                                                                                                                          19.4, 19.6, 17.7, 21.3, 22.1, 17.9, 28.2, 16.3, 25.3, 19.7, 21.7, 
                                                                                                                                                          19, 18.8, 11.8, 15.6, 9.8, 17.7)), .Names = c("x", "y", "z", 
                                                                                                                                                                                                        "soil_m_sat"), class = "data.frame", row.names = c(NA, -296L))
</code></pre>

<p>In order to estimate a variogram for this data I need to remove the spatial trend from it. The soil moisture, of course, varies with the surface - the higher a point is the dryer it is. And since this soil moisture data is percetagewise the relationship is hardly linear, what leads me to allow up to cubic dependencies of the soil moisture to the z-coordinate. It happens that in this area there is a small more or less elliptic elevation, so that I want to allow the soil moisture to be dependend of the x- and y-coordinates in a quadratic way. I hope the following model does exactly this:</p>

<pre><code>polymod &lt;- lm(soil_m_sat ~ poly(x + y, degree = 2) + poly(z, degree = 3), data = gue)
summary(polymod)
</code></pre>

<p>The summary shows me that there is no significance for the first coefficient of the x- and y-dependency (what summary names <code>poly(x + y, degree = 2)1</code>). Because the help page from <code>poly()</code> told me that it "returns or evaluates orthogonal polynomials of degree 1 to <code>degree</code>", I thought, removing a degree one polynom from the model might be the same as removing the first coefficient of the degree 2 polynom. Therefore I tried to remove it like this:</p>

<pre><code>mod &lt;- lm(soil_m_sat ~ poly(x + y, degree = 2) - poly(x + y, degree = 1) + poly(z, degree = 3), data = gue)
summary(mod)
</code></pre>

<p>But the summary of mod looks exactly the same as the summary of polymod, meaning <code>mod</code> does not differ from <code>polymod</code>. How is it possible to remove the unsignificant component then?</p>