R：如何或应该在线性模型中删除无关紧要的正交多项式基？
R: How to or should I drop an insignificant orthogonal polynomial basis in a linear model?
我有 x、y 和 z 坐标的土壤水分数据，如下所示：
gue <- 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))
为了估计此数据的变异函数，我需要从中删除空间趋势。当然，土壤水分随表面变化——越高越干燥。由于此土壤水分数据是按百分比计算的，因此关系几乎不是线性的，这导致我允许土壤水分与 z 坐标的立方依赖关系。碰巧在这个区域有一个或多或少的椭圆形高程，所以我想让土壤水分以二次方式依赖于 x 和 y 坐标。我希望下面的 model 正是这样做的：
polymod <- lm(soil_m_sat ~ poly(x + y, degree = 2) + poly(z, degree = 3), data = gue)
summary(polymod)
摘要显示 x 和 y 相关性的第一个系数没有意义（摘要名称 poly(x + y, degree = 2)1）。因为 poly() 的帮助页面告诉我它 "returns or evaluates orthogonal polynomials of degree 1 to degree"，我想，从 model 中删除一个一次多项式可能与删除二次多项式的第一个系数相同.因此我试着像这样删除它：
mod <- lm(soil_m_sat ~ poly(x + y, degree = 2) - poly(x + y, degree = 1) + poly(z, degree = 3), data = gue)
summary(mod)
但是 mod 的摘要看起来与 polymod 的摘要完全一样，意思是 mod 与 polymod 没有区别。那不重要的成分怎么可能去掉呢？
不，在这种情况下不要用 summary 检查。你应该使用 anova。 poly() 中的多项式项或 bs() 中的样条项包含的系数不止一个，因此它们更像是具有多个水平的因子变量。
> anova(polymod)
Analysis of Variance Table
Response: soil_m_sat
Df Sum Sq Mean Sq F value Pr(>F)
poly(x + y, degree = 2) 2 113484 56742 1600.8 < 2.2e-16 ***
poly(z, degree = 3) 3 68538 22846 644.5 < 2.2e-16 ***
Residuals 290 10280 35
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
ANOVA table 清楚地表明您需要所有模型项。不掉任何东西。
但我还是要回答你的问题，让你开心。
删除 poly(x + y, degree = 2)1 项并非不可能，但您需要为此目的访问模型矩阵。你可以
gue$XY_poly <- with(gue, poly(x + y, degree = 2))[, 2] ## use the 2nd column only
fit <- lm(soil_m_sat ~ XY_poly + poly(z, degree = 3), data = gue)
summary(fit)
## ...
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 52.3071 0.3459 151.217 < 2e-16 ***
XY_poly -18.8515 7.3894 -2.551 0.0112 *
poly(z, degree = 3)1 -418.1634 6.4937 -64.395 < 2e-16 ***
poly(z, degree = 3)2 116.5327 6.9171 16.847 < 2e-16 ***
poly(z, degree = 3)3 -28.7773 5.9517 -4.835 2.16e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 5.951 on 291 degrees of freedom
Multiple R-squared: 0.9464, Adjusted R-squared: 0.9457
F-statistic: 1285 on 4 and 291 DF, p-value: < 2.2e-16
我有 x、y 和 z 坐标的土壤水分数据，如下所示：
gue <- 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,
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25.332, 25.333, 25.333, 25.346, 25.344, 25.339, 25.355, 25.362,
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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,
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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,
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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,
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42.6, 46, 43.8, 45.4, 42.2, 46.5, 40.4, 39.9, 53, 44.7, 35.8,
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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))
为了估计此数据的变异函数，我需要从中删除空间趋势。当然，土壤水分随表面变化——越高越干燥。由于此土壤水分数据是按百分比计算的，因此关系几乎不是线性的，这导致我允许土壤水分与 z 坐标的立方依赖关系。碰巧在这个区域有一个或多或少的椭圆形高程，所以我想让土壤水分以二次方式依赖于 x 和 y 坐标。我希望下面的 model 正是这样做的：
polymod <- lm(soil_m_sat ~ poly(x + y, degree = 2) + poly(z, degree = 3), data = gue)
summary(polymod)
摘要显示 x 和 y 相关性的第一个系数没有意义（摘要名称 poly(x + y, degree = 2)1）。因为 poly() 的帮助页面告诉我它 "returns or evaluates orthogonal polynomials of degree 1 to degree"，我想，从 model 中删除一个一次多项式可能与删除二次多项式的第一个系数相同.因此我试着像这样删除它：
mod <- lm(soil_m_sat ~ poly(x + y, degree = 2) - poly(x + y, degree = 1) + poly(z, degree = 3), data = gue)
summary(mod)
但是 mod 的摘要看起来与 polymod 的摘要完全一样，意思是 mod 与 polymod 没有区别。那不重要的成分怎么可能去掉呢？
不，在这种情况下不要用 summary 检查。你应该使用 anova。 poly() 中的多项式项或 bs() 中的样条项包含的系数不止一个，因此它们更像是具有多个水平的因子变量。
> anova(polymod)
Analysis of Variance Table
Response: soil_m_sat
Df Sum Sq Mean Sq F value Pr(>F)
poly(x + y, degree = 2) 2 113484 56742 1600.8 < 2.2e-16 ***
poly(z, degree = 3) 3 68538 22846 644.5 < 2.2e-16 ***
Residuals 290 10280 35
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
ANOVA table 清楚地表明您需要所有模型项。不掉任何东西。
但我还是要回答你的问题，让你开心。
删除 poly(x + y, degree = 2)1 项并非不可能，但您需要为此目的访问模型矩阵。你可以
gue$XY_poly <- with(gue, poly(x + y, degree = 2))[, 2] ## use the 2nd column only
fit <- lm(soil_m_sat ~ XY_poly + poly(z, degree = 3), data = gue)
summary(fit)
## ...
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 52.3071 0.3459 151.217 < 2e-16 ***
XY_poly -18.8515 7.3894 -2.551 0.0112 *
poly(z, degree = 3)1 -418.1634 6.4937 -64.395 < 2e-16 ***
poly(z, degree = 3)2 116.5327 6.9171 16.847 < 2e-16 ***
poly(z, degree = 3)3 -28.7773 5.9517 -4.835 2.16e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 5.951 on 291 degrees of freedom
Multiple R-squared: 0.9464, Adjusted R-squared: 0.9457
F-statistic: 1285 on 4 and 291 DF, p-value: < 2.2e-16