library(MASS) library(dplyr) library(magrittr) library(ggplot2) library(mice) library(DAAG) library(car)
Statistical Programming with R
library(MASS) library(dplyr) library(magrittr) library(ggplot2) library(mice) library(DAAG) library(car)
Linear regression model \[ y_i=\alpha+\beta{x}_i+\varepsilon_i \]
Assumptions:
anscombe %>% ggplot(aes(y1, x1)) + geom_point() + geom_smooth(method = "lm")
The linear model:
fit <- data %>% lm(y ~ x) summary(fit)
Output:
boys
examplefit <- anscombe %$% lm(y1 ~ x1) summary(fit)
## ## Call: ## lm(formula = y1 ~ x1) ## ## Residuals: ## Min 1Q Median 3Q Max ## -1.92127 -0.45577 -0.04136 0.70941 1.83882 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 3.0001 1.1247 2.667 0.02573 * ## x1 0.5001 0.1179 4.241 0.00217 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 1.237 on 9 degrees of freedom ## Multiple R-squared: 0.6665, Adjusted R-squared: 0.6295 ## F-statistic: 17.99 on 1 and 9 DF, p-value: 0.00217
qqnorm()
qqnorm
with simulated errors cf. rnorm(n, 0, s)
anscombe %>% ggplot(aes(x1, y1)) + geom_point() + geom_smooth(method = "loess", col = "blue") + geom_smooth(method = "lm", col = "orange")
The loess curve suggests slight non-linearity
anscombe %$% lm(y1 ~ x1 + I(x1^2)) %>% summary()
## ## Call: ## lm(formula = y1 ~ x1 + I(x1^2)) ## ## Residuals: ## Min 1Q Median 3Q Max ## -1.8704 -0.3481 -0.2456 0.7129 1.8072 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 0.75507 3.28815 0.230 0.824 ## x1 1.06925 0.78982 1.354 0.213 ## I(x1^2) -0.03162 0.04336 -0.729 0.487 ## ## Residual standard error: 1.27 on 8 degrees of freedom ## Multiple R-squared: 0.6873, Adjusted R-squared: 0.6092 ## F-statistic: 8.793 on 2 and 8 DF, p-value: 0.009558
par(mfrow = c(1, 2)) fit %>% plot(which = c(1, 3), cex = .6)
par(mfrow = c(1, 2)) boys %$% lm(bmi ~ age) %>% plot(which = c(1, 3), cex = .6)
fit %>% plot(which = 2, cex = .6)
The QQplot shows some divergence from normality at the tails
Leverage: see the fit line as a lever.
Standardized residuals:
Cook’s distance: how far the predicted values would move if your model were fit without the data point in question.
Outliers are cases with large \(e_z\) (standardized residuals).
If the model is correct we expect:
Influential cases are cases with large influence on parameter estimates
par(mfrow = c(1, 2), cex = .6) fit %>% plot(which = c(4, 5))
There are no cases with \(|e_z|>2\), so no outliers (right plot). There are no cases with Cook’s Distance \(>1\), but case 3 stands out
boys.fit <- na.omit(boys) %$% # Extremely wasteful lm(age ~ reg)
boys.fit %>% anova()
## Analysis of Variance Table ## ## Response: age ## Df Sum Sq Mean Sq F value Pr(>F) ## reg 4 14.7 3.6747 0.3701 0.8298 ## Residuals 218 2164.6 9.9293
boys.fit %>% summary()
## ## Call: ## lm(formula = age ~ reg) ## ## Residuals: ## Min 1Q Median 3Q Max ## -5.8519 -2.5301 0.0254 2.2274 6.2614 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 14.7109 0.5660 25.993 <2e-16 *** ## regeast -0.8168 0.7150 -1.142 0.255 ## regwest -0.7044 0.6970 -1.011 0.313 ## regsouth -0.6913 0.6970 -0.992 0.322 ## regcity -0.6663 0.9038 -0.737 0.462 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 3.151 on 218 degrees of freedom ## Multiple R-squared: 0.006745, Adjusted R-squared: -0.01148 ## F-statistic: 0.3701 on 4 and 218 DF, p-value: 0.8298
boys.fit %>% model.matrix() %>% head()
## (Intercept) regeast regwest regsouth regcity ## 1 1 0 0 0 0 ## 2 1 0 0 1 0 ## 3 1 0 0 1 0 ## 4 1 1 0 0 0 ## 5 1 0 1 0 0 ## 6 1 1 0 0 0
aov()
boys.fit %>% aov()
## Call: ## aov(formula = .) ## ## Terms: ## reg Residuals ## Sum of Squares 14.6987 2164.5869 ## Deg. of Freedom 4 218 ## ## Residual standard error: 3.151079 ## Estimated effects may be unbalanced
aov()
is for balanced designs.
anova()
boys.fit %>% anova()
## Analysis of Variance Table ## ## Response: age ## Df Sum Sq Mean Sq F value Pr(>F) ## reg 4 14.7 3.6747 0.3701 0.8298 ## Residuals 218 2164.6 9.9293
Anova()
with type I Sum of SquaresAnova()
with type II Sum of SquaresNo interaction assumed. So test for this first!
Anova()
with type II Sum of Squaresboys.fit %>% car::Anova(type = 2)
## Anova Table (Type II tests) ## ## Response: age ## Sum Sq Df F value Pr(>F) ## reg 14.7 4 0.3701 0.8298 ## Residuals 2164.6 218
Anova()
with type III Sum of SquaresTests if there is a main effect, given the other main effect and the interaction.
When data are balanced, Type I, II and II are identical, because the factors are orthogonal
Anova()
with type III Sum of Squaresboys.fit %>% car::Anova(type = 3)
## Anova Table (Type III tests) ## ## Response: age ## Sum Sq Df F value Pr(>F) ## (Intercept) 6708.8 1 675.6530 <2e-16 *** ## reg 14.7 4 0.3701 0.8298 ## Residuals 2164.6 218 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
coef <- boys.fit %>% aov() %>% summary.lm() coef
## ## Call: ## aov(formula = .) ## ## Residuals: ## Min 1Q Median 3Q Max ## -5.8519 -2.5301 0.0254 2.2274 6.2614 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 14.7109 0.5660 25.993 <2e-16 *** ## regeast -0.8168 0.7150 -1.142 0.255 ## regwest -0.7044 0.6970 -1.011 0.313 ## regsouth -0.6913 0.6970 -0.992 0.322 ## regcity -0.6663 0.9038 -0.737 0.462 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 3.151 on 218 degrees of freedom ## Multiple R-squared: 0.006745, Adjusted R-squared: -0.01148 ## F-statistic: 0.3701 on 4 and 218 DF, p-value: 0.8298
p.val <- coef$coefficients p.adjust(p.val[, "Pr(>|t|)"], method = "bonferroni")
## (Intercept) regeast regwest regsouth regcity ## 5.077098e-68 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
Akaike’s An Information Criterion
boys.fit %>% AIC()
## [1] 1151.684
and Bayesian Information Criterion
boys.fit %>% BIC()
## [1] 1172.127
boys.fit2 <- na.omit(boys) %$% lm(age ~ reg + hgt) boys.fit %>% AIC()
## [1] 1151.684
boys.fit2 %>% AIC()
## [1] 836.3545
boys.fit3 <- na.omit(boys) %$% lm(age ~ reg + hgt * wgt)
is equivalent to
boys.fit3 <- na.omit(boys) %$% lm(age ~ reg + hgt + wgt + hgt:wgt)
boys.fit %>% AIC()
## [1] 1151.684
boys.fit2 %>% AIC()
## [1] 836.3545
boys.fit3 %>% AIC()
## [1] 823.0386
anova(boys.fit, boys.fit2, boys.fit3)
## Analysis of Variance Table ## ## Model 1: age ~ reg ## Model 2: age ~ reg + hgt ## Model 3: age ~ reg + hgt * wgt ## Res.Df RSS Df Sum of Sq F Pr(>F) ## 1 218 2164.59 ## 2 217 521.64 1 1642.94 731.83 < 2.2e-16 *** ## 3 215 482.67 2 38.97 8.68 0.000237 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
boys.fit3 %>% car::Anova(type = 3)
## Anova Table (Type III tests) ## ## Response: age ## Sum Sq Df F value Pr(>F) ## (Intercept) 27.04 1 12.0434 0.000628 *** ## reg 5.90 4 0.6569 0.622651 ## hgt 106.79 1 47.5667 5.84e-11 *** ## wgt 7.33 1 3.2654 0.072152 . ## hgt:wgt 2.99 1 1.3324 0.249655 ## Residuals 482.67 215 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
boys.fit3 %>% car::Anova(type = 2)
## Anova Table (Type II tests) ## ## Response: age ## Sum Sq Df F value Pr(>F) ## reg 5.90 4 0.6569 0.6227 ## hgt 170.35 1 75.8810 8.114e-16 *** ## wgt 35.98 1 16.0276 8.595e-05 *** ## hgt:wgt 2.99 1 1.3324 0.2497 ## Residuals 482.67 215 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
boys.fit3 %>% anova()
## Analysis of Variance Table ## ## Response: age ## Df Sum Sq Mean Sq F value Pr(>F) ## reg 4 14.70 3.67 1.6368 0.1661 ## hgt 1 1642.94 1642.94 731.8311 < 2.2e-16 *** ## wgt 1 35.98 35.98 16.0276 8.595e-05 *** ## hgt:wgt 1 2.99 2.99 1.3324 0.2497 ## Residuals 215 482.67 2.24 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
DfBeta calculates the change in coefficients depicted as deviation in SE’s.
boys.fit3 %>% dfbeta()
## (Intercept) regeast regwest regsouth regcity ## 1 0.3044652932 -3.264520e-02 -3.261652e-02 -3.044991e-02 -3.182952e-02 ## 2 -0.1235795768 3.358160e-03 3.189464e-03 -3.554200e-02 5.056504e-03 ## 3 -0.1752265093 1.479115e-03 1.511858e-03 -1.344283e-02 1.493751e-04 ## 4 -0.0885783023 -6.195476e-03 7.649256e-04 2.461411e-04 -1.545015e-04 ## 5 0.3171577553 3.385884e-03 -5.723827e-02 3.214539e-03 1.429112e-02 ## 6 -0.1755467631 -1.894555e-02 2.433447e-03 1.512673e-03 6.986407e-04 ## 7 -0.0820077163 1.375353e-02 1.376671e-02 1.343853e-02 1.311841e-02 ## 8 0.6567154608 -2.417240e-03 2.144013e-02 1.443837e-03 4.591409e-03 ## 9 -0.1359593587 1.658664e-03 1.600995e-03 -1.816962e-02 1.843015e-03 ## 10 -0.1354497538 1.942417e-03 -1.478265e-02 1.073443e-03 1.236205e-03 ## 11 -0.1039371719 2.504727e-02 2.503147e-02 2.464443e-02 2.467985e-02 ## 12 -0.1628699441 3.490049e-03 3.532404e-03 -2.383319e-02 1.749193e-03 ## 13 0.1815161501 -9.762117e-04 -1.040323e-03 9.387137e-03 7.873466e-04 ## 14 0.0893668872 -5.518048e-04 -5.805184e-04 5.124512e-03 2.828278e-04 ## 15 -0.0695126584 3.263701e-03 -2.650946e-02 2.667489e-03 4.166140e-03 ## 16 0.0248305784 1.778517e-03 -2.118963e-04 -6.212807e-05 3.042458e-05 ## 17 -0.1257621945 2.120998e-03 -1.435104e-02 1.486645e-03 1.086780e-03 ## 18 -0.1194293254 -1.737389e-02 2.329707e-03 1.820923e-03 1.063632e-03 ## 19 -0.0681221241 -6.129848e-03 8.199909e-04 4.668188e-04 6.260524e-05 ## 20 -0.0694280001 7.684668e-04 -6.017549e-03 3.144575e-04 3.476453e-04 ## 21 -0.0943586044 -1.320165e-02 1.420288e-03 8.033362e-04 1.021722e-03 ## 22 0.1439909585 -8.707260e-04 -9.128643e-04 8.383771e-03 4.070590e-04 ## 23 -0.0834099137 1.156304e-03 -9.225201e-03 5.771735e-04 8.331653e-04 ## 24 -0.0435156593 4.134683e-04 4.264501e-04 -3.307434e-03 7.673603e-07 ## 25 -0.0683512055 1.049958e-03 -7.092767e-03 7.008062e-04 4.626599e-04 ## 26 -0.0160960201 2.995715e-03 -2.360159e-02 2.877161e-03 3.972212e-03 ## 27 0.0524526917 -4.416041e-04 -4.441380e-04 4.423547e-03 -1.568658e-04 ## 28 0.3487348866 -2.075991e-03 -2.194659e-03 1.925724e-02 1.272359e-03 ## 29 0.0459496743 -4.436352e-04 -4.510420e-04 3.901929e-03 -1.048794e-04 ## 30 -0.0794974274 1.000304e-03 9.286279e-04 8.634369e-05 -4.623303e-02 ## 31 0.0978065719 -8.925021e-04 7.941297e-03 -1.607611e-04 -4.944141e-04 ## 32 -0.0280150956 -1.723636e-02 2.117584e-03 2.158896e-03 2.059924e-03 ## 33 0.1673085657 4.789079e-03 -3.529753e-02 6.496940e-03 6.482271e-03 ## 34 0.0222336673 -2.218578e-02 1.756702e-03 1.704978e-03 3.760054e-03 ## 35 -0.0410939875 4.374034e-04 4.477695e-04 2.076306e-04 -1.330940e-02 ## 36 -0.0275456313 1.594643e-03 1.527860e-03 -1.541401e-02 2.299745e-03 ## 37 -0.0009810534 -1.488133e-02 1.151765e-03 9.602692e-04 2.487377e-03 ## 38 -0.0037731939 -1.881191e-02 2.135209e-03 2.293426e-03 2.550659e-03 ## 39 0.0233852789 3.760318e-03 -3.845430e-04 -2.206292e-04 -3.462361e-04 ## 40 0.0424737750 5.568379e-03 -7.667067e-04 -5.862484e-04 -2.806670e-04 ## 41 -0.0087620636 2.060169e-04 2.052740e-04 -1.601251e-03 1.566422e-04 ## 42 0.0229176059 1.089386e-03 9.167080e-04 5.535242e-04 -6.762522e-02 ## 43 -0.0004230876 -1.453827e-04 1.447272e-05 1.139613e-05 1.832953e-05 ## 44 0.2465082772 -2.465479e-03 1.855938e-02 -9.583646e-04 -5.734926e-04 ## 45 0.0154770933 -1.522299e-02 1.470594e-03 1.604720e-03 2.342361e-03 ## 46 0.4000567139 -3.097280e-03 2.529598e-02 -4.966661e-04 -1.387541e-04 ## 47 0.0129251286 1.880304e-02 1.870603e-02 1.877882e-02 2.022359e-02 ## 48 0.1409420495 1.808927e-02 -1.871803e-03 -9.002755e-04 -1.333152e-03 ## 49 0.5927112051 -4.407173e-03 3.003008e-02 -1.266998e-03 2.089692e-03 ## 50 -0.0019510643 6.667981e-04 6.132732e-04 -7.916248e-03 1.392733e-03 ## 51 0.1850793140 3.259784e-03 3.134372e-03 -3.312223e-02 5.350301e-03 ## 52 0.0401694562 3.494507e-04 1.891399e-04 -1.096689e-02 2.929103e-03 ## 53 0.0250810809 2.311251e-02 2.300402e-02 2.322896e-02 2.471696e-02 ## 54 0.1585843048 2.912889e-03 -2.717772e-02 4.076413e-03 5.227141e-03 ## 55 0.0053021596 -3.916413e-03 2.745975e-04 2.534731e-04 7.072498e-04 ## 56 0.0510981498 1.321773e-03 -1.378784e-02 1.459278e-03 2.890028e-03 ## 57 0.0009346512 1.074564e-03 -1.108857e-04 -1.076724e-04 -1.487941e-04 ## 58 0.0104006919 1.218211e-03 1.157519e-03 1.201568e-03 -5.019055e-02 ## 59 0.0019607875 9.818199e-04 9.717943e-04 -6.939773e-03 9.963548e-04 ## 60 0.0015909046 -1.589076e-04 9.974048e-04 -1.695701e-04 -1.407230e-04 ## 61 0.0866900084 4.135805e-02 4.122922e-02 4.233469e-02 4.331619e-02 ## 62 0.1377240791 -1.667497e-03 -1.727126e-03 -1.068759e-03 4.467279e-02 ## 63 0.0142493376 -8.047863e-04 5.816638e-03 -7.476932e-04 -8.526886e-04 ## 64 0.0100491747 9.533773e-04 9.017921e-04 9.351524e-04 -4.026844e-02 ## 65 0.5919299685 -4.133753e-03 -4.241026e-03 4.089957e-02 1.760366e-04 ## 66 0.0120546062 8.527751e-04 7.948724e-04 7.999417e-04 -3.865405e-02 ## 67 0.0511934790 1.512372e-02 -1.670354e-03 -1.396954e-03 -1.709988e-03 ## 68 0.0099729328 -8.115067e-04 4.902479e-03 -8.683153e-04 -6.584361e-04 ## 69 0.0074002447 -7.633362e-04 4.400424e-03 -8.473035e-04 -5.889415e-04 ## 70 0.0730638905 1.282627e-03 -1.380631e-02 1.641636e-03 2.835433e-03 ## 71 0.0080853933 -1.076903e-03 6.978927e-03 -1.150638e-03 -1.027369e-03 ## 72 0.0367823175 7.598085e-04 -7.434360e-03 9.738705e-04 1.487783e-03 ## 73 0.0137526936 -1.585720e-03 1.504692e-02 -1.223719e-03 -2.810625e-03 ## 74 0.1006099208 1.631588e-03 -1.535421e-02 2.420689e-03 2.873993e-03 ## 75 0.2114052216 -6.925523e-04 -1.069861e-03 -2.584290e-02 6.005616e-03 ## 76 0.0976568576 3.209633e-02 3.193474e-02 3.265297e-02 3.480178e-02 ## 77 0.2355073379 1.002101e-03 8.899275e-04 -3.420667e-02 3.399089e-03 ## 78 0.0647096206 1.391528e-03 1.366178e-03 2.080124e-03 -5.415685e-02 ## 79 0.1699244705 -2.522319e-02 8.568072e-04 1.879289e-03 3.789134e-03 ## 80 0.1372024762 6.498668e-04 4.978292e-04 -2.110098e-02 3.385823e-03 ## 81 -0.0125634441 -5.788862e-04 4.393240e-03 -7.022455e-04 -8.006755e-04 ## 82 0.2828841464 -1.130475e-02 -1.184098e-02 -7.566412e-02 -2.870580e-04 ## 83 0.2774242422 5.699008e-04 -3.484489e-02 2.244584e-03 3.990396e-03 ## 84 0.1851997139 4.594739e-04 3.416915e-04 -2.756013e-02 2.854366e-03 ## 85 -0.0162206530 -6.543197e-04 -6.303879e-04 5.913803e-03 -9.833366e-04 ## 86 0.1372366476 3.402687e-02 -4.891884e-03 -4.752249e-03 -2.928489e-03 ## 87 -0.0311053345 8.505307e-03 -7.346638e-04 -9.690687e-04 -1.405340e-03 ## 88 0.0380901246 3.525314e-04 3.020631e-04 4.808851e-04 -2.753269e-02 ## 89 0.2145430561 -2.874261e-02 -9.382930e-04 -3.284281e-04 4.293208e-03 ## 90 0.2924833945 -2.760917e-03 -4.940153e-02 -1.480488e-03 1.002419e-03 ## 91 0.3196307525 -5.459140e-02 8.366140e-05 3.259953e-03 -8.864101e-05 ## 92 -0.0347835228 -1.034866e-03 8.536026e-03 -1.313401e-03 -1.621054e-03 ## 93 0.1673040775 -2.407752e-02 6.176385e-04 1.762584e-03 2.845773e-03 ## 94 -0.0440341033 -3.976356e-02 -3.963124e-02 -4.032592e-02 -4.162749e-02 ## 95 -0.0067365553 -4.538955e-04 -4.607804e-04 -6.330279e-04 1.089624e-02 ## 96 0.0257504174 -4.007729e-03 4.428681e-04 7.206889e-04 5.831780e-04 ## 97 -0.0400330467 7.894717e-03 -8.283890e-04 -1.238974e-03 -1.205778e-03 ## 98 -0.0496415394 -6.712238e-04 8.166237e-03 -9.024787e-04 -1.688996e-03 ## 99 0.2610880219 -3.890382e-02 9.338437e-04 3.594228e-03 1.012634e-03 ## 100 -0.0262220278 -2.431649e-03 -2.322674e-03 2.326781e-02 -3.850323e-03 ## 101 0.1043252985 -8.617000e-03 -8.709913e-03 -9.750462e-03 -1.630997e-01 ## 102 0.0548718632 -7.583228e-03 6.262388e-04 1.158633e-03 9.819731e-04 ## 103 -0.0483035191 2.844421e-02 -3.613460e-03 -4.506267e-03 -3.963377e-03 ## 104 -0.0722189000 -3.121159e-02 -3.111901e-02 -3.203143e-02 -3.264270e-02 ## 105 0.1391349953 1.415790e-03 1.435396e-03 -1.577201e-02 1.370683e-03 ## 106 0.1014188178 3.200318e-02 3.195267e-02 3.317967e-02 3.300534e-02 ## 107 0.2264582827 -1.490978e-03 -4.320065e-02 8.167269e-04 -2.119195e-03 ## 108 0.3090263673 -1.261237e-01 -1.258726e-01 -1.249816e-01 -1.277516e-01 ## 109 -0.0863279553 -6.637342e-04 1.237721e-02 -8.817348e-04 -2.777342e-03 ## 110 0.0696877641 1.835997e-04 1.428785e-04 -1.035292e-02 1.027125e-03 ## 111 0.0648637181 2.053934e-02 2.046729e-02 2.100816e-02 2.186110e-02 ## 112 0.0874653713 -1.196810e-02 5.871399e-04 1.375436e-03 1.162399e-03 ## 113 -0.0743099053 -1.946088e-03 -1.867003e-03 1.863052e-02 -3.136516e-03 ## 114 0.0622123056 1.597845e-04 1.283990e-04 -9.252507e-03 8.361159e-04 ## 115 -0.0428414456 6.403799e-03 -3.999684e-04 -7.444708e-04 -9.044305e-04 ## 116 -0.0385219223 -1.723950e-03 -1.587112e-03 -1.635584e-03 8.335675e-02 ## 117 0.0591727358 6.373183e-05 3.111515e-05 -9.003491e-03 7.633776e-04 ## 118 -0.0203964515 -1.855475e-04 -1.732317e-04 -3.334380e-04 1.283683e-02 ## 119 -0.0132454180 -4.731580e-05 1.631997e-03 -1.107523e-04 -2.654537e-04 ## 120 -0.1065340596 -5.865019e-04 1.374922e-02 -9.702948e-04 -2.779765e-03 ## 121 0.0250447422 -7.315139e-03 -5.109141e-02 -7.720801e-03 -8.115720e-03 ## 122 0.0802390973 3.246566e-02 3.234742e-02 3.274208e-02 3.476525e-02 ## 123 0.1035188462 7.377409e-04 -1.149656e-02 1.790909e-03 9.773521e-04 ## 124 0.0258914714 -1.220101e-04 -1.513221e-04 -3.932444e-03 4.408914e-04 ## 125 0.1311222001 -4.959153e-05 5.301903e-05 -2.331117e-02 -1.192938e-03 ## 126 -0.0372280745 -2.771661e-04 -2.596173e-04 5.291758e-03 -6.352937e-04 ## 127 -0.0774308819 1.081326e-02 -7.608933e-04 -1.467398e-03 -1.368861e-03 ## 128 0.0293588067 1.037388e-02 1.034632e-02 1.060973e-02 1.090910e-02 ## 129 0.1134847023 -2.973853e-02 -1.856801e-03 -1.166653e-03 -1.258179e-03 ## 130 0.0593085589 -6.575205e-04 -7.043218e-04 -1.137129e-02 3.722824e-04 ## 131 0.0670679826 -5.368267e-04 -9.039102e-03 -4.317653e-04 9.029282e-04 ## 132 0.0652794579 -1.420762e-04 -1.382125e-04 -1.149718e-02 4.153221e-05 ## 133 0.1258673466 -4.921736e-02 -9.080199e-03 -1.085463e-02 -2.432891e-03 ## 134 0.0544498214 3.401475e-02 3.402029e-02 3.483483e-02 3.422589e-02 ## 135 -0.6574660738 -1.955048e-02 -1.888692e-02 -8.239636e-02 -2.916813e-02 ## 136 0.0647702702 4.401583e-02 4.410132e-02 4.552628e-02 4.298277e-02 ## 137 0.0535274237 2.115135e-04 2.241443e-04 -7.651461e-03 1.652443e-04 ## 138 0.0794209106 -3.243550e-02 -3.686027e-03 -3.803039e-03 -2.252711e-03 ## 139 0.0431201088 -1.140883e-04 -1.145704e-04 -7.639946e-03 5.748419e-05 ## 140 0.0320901299 -4.736753e-03 1.086846e-04 4.107003e-04 2.116895e-04 ## 141 -0.4541663597 -1.598013e-02 -6.744434e-02 -1.990669e-02 -2.370928e-02 ## 142 0.0427541764 -7.839789e-03 -2.754769e-04 -1.417859e-05 1.775059e-04 ## 143 -0.0527689454 7.612545e-03 -1.928135e-04 -6.352981e-04 -5.795375e-04 ## 144 -0.1871375277 5.901132e-02 5.924225e-02 5.823828e-02 5.577734e-02 ## 145 -0.3520429205 7.099594e-02 7.151275e-02 7.016703e-02 6.301346e-02 ## 146 -0.0818548755 -4.644805e-04 -4.404372e-04 1.147479e-02 -1.047409e-03 ## 147 -0.1554600826 -4.824905e-02 -4.815072e-02 -5.004706e-02 -5.000438e-02 ## 148 0.0318882967 1.655483e-02 1.657205e-02 1.712664e-02 1.640844e-02 ## 149 -0.0274041387 -3.929582e-03 -3.722565e-03 -3.760364e-03 -9.059248e-02 ## 150 -0.0687296355 -3.755180e-04 -3.681922e-04 9.487128e-03 -6.669328e-04 ## 151 0.0502510094 -2.756412e-02 -1.831007e-03 -1.117942e-03 -3.696299e-03 ## 152 -0.0495477223 1.169680e-04 1.417645e-04 8.145795e-03 -4.554389e-04 ## 153 0.0229330642 -1.808292e-04 -3.868199e-03 -5.368341e-05 5.054196e-05 ## 154 -0.0198379464 -5.886644e-07 -4.940160e-07 3.231572e-03 -6.953136e-05 ## 155 0.0014496101 -1.834071e-05 -1.854861e-05 -3.326861e-04 -7.398965e-06 ## 156 -0.0891400285 1.088724e-04 1.577730e-04 -3.980382e-04 4.969357e-02 ## 157 -0.0105078627 4.025873e-05 3.637481e-05 -6.685020e-05 6.877775e-03 ## 158 -0.0006254243 -3.551472e-04 -3.550179e-04 -3.632040e-04 -3.604559e-04 ## 159 -0.1621291456 -4.718260e-02 -4.694284e-02 -4.788468e-02 -5.149926e-02 ## 160 0.0156887901 2.193972e-02 2.200618e-02 2.263198e-02 2.103522e-02 ## 161 0.0084459749 -2.444143e-02 -2.245781e-03 -1.984676e-03 -4.168515e-03 ## 162 0.0420668686 -2.703768e-04 -8.447148e-03 2.406060e-04 -6.209853e-04 ## 163 0.0101325197 -1.408526e-03 -1.337246e-03 -1.408767e-02 -2.271023e-03 ## 164 0.0120513500 -2.468841e-03 -2.230038e-02 -2.054906e-03 -4.216733e-03 ## 165 -1.9576644651 8.945670e-02 9.097862e-02 7.548530e-02 6.446316e-02 ## 166 0.0182528899 -1.344731e-04 -1.066865e-04 -4.526972e-03 -4.845755e-04 ## 167 -0.0007047179 -2.787001e-03 -1.921726e-02 -2.928300e-03 -3.426282e-03 ## 168 -0.0687503938 -8.529272e-05 9.389579e-03 -7.171690e-04 -2.896711e-04 ## 169 -0.1381682494 4.026651e-04 4.995050e-04 2.228904e-02 -1.636189e-03 ## 170 -0.1457982212 -4.544570e-04 -4.200364e-04 2.146202e-02 -1.418204e-03 ## 171 -0.0943608906 -3.336838e-03 -2.922652e-03 -2.854230e-02 -9.602219e-03 ## 172 -0.0580173879 7.734197e-04 8.019094e-04 1.263888e-02 1.557980e-05 ## 173 0.0155906934 -3.649154e-04 -3.101532e-04 -6.732093e-03 -1.109233e-03 ## 174 -0.0332099120 4.198734e-04 6.675188e-03 2.271172e-04 1.677028e-04 ## 175 -0.0222536159 3.761945e-04 5.354096e-03 2.175724e-04 3.266742e-04 ## 176 -0.5907061469 -1.315553e-02 -1.238267e-02 -1.658779e-02 -1.414766e-01 ## 177 -0.0121932400 1.127472e-02 1.132533e-02 1.148068e-02 1.053790e-02 ## 178 -0.0078454368 -1.623253e-03 -1.059138e-02 -1.740312e-03 -2.110706e-03 ## 179 -0.0370450031 4.173463e-04 8.317848e-03 3.397184e-05 6.007368e-04 ## 180 -0.0506447147 1.466123e-02 1.073774e-03 8.162924e-04 7.454833e-04 ## 181 -0.0191188165 6.472674e-04 6.129119e-04 8.555294e-03 1.019271e-03 ## 182 -0.0111352046 4.711726e-04 4.916055e-03 3.672910e-04 6.207965e-04 ## 183 -0.0110372662 1.048148e-02 1.348923e-03 1.470239e-03 1.201709e-03 ## 184 -0.0356997231 1.368846e-02 8.386538e-04 4.629185e-04 1.403953e-03 ## 185 -0.1966761861 3.648152e-02 3.427186e-03 3.260958e-03 -1.482172e-03 ## 186 -0.2957464117 -9.485206e-02 -9.466077e-02 -9.798516e-02 -9.852996e-02 ## 187 -0.0019828333 6.622138e-04 5.779833e-03 5.687379e-04 1.116771e-03 ## 188 -0.0915041799 -5.621642e-02 -5.607028e-02 -5.642459e-02 -5.927447e-02 ## 189 -0.0222900161 2.299702e-03 9.897269e-03 2.929764e-03 6.916240e-04 ## 190 -0.1564166076 -3.709740e-03 -1.826434e-02 -4.192504e-03 -8.209076e-03 ## 191 0.0496303703 -2.478918e-02 -2.478241e-02 -2.405059e-02 -2.524264e-02 ## 192 -0.2927749800 2.143719e-03 4.345874e-02 1.148857e-03 -2.674958e-03 ## 193 -0.3182627009 6.448766e-04 8.377050e-04 5.114639e-02 -3.538135e-03 ## 194 -0.1657157389 2.181009e-03 2.177352e-03 3.965770e-02 1.344345e-03 ## 195 0.0545623651 2.290859e-03 2.191127e-03 1.324373e-02 3.683401e-03 ## 196 0.0168043053 1.340196e-02 1.120834e-03 8.632711e-04 3.367276e-03 ## 197 -0.3209959479 -8.340988e-04 -7.742932e-04 -3.511386e-03 1.802757e-01 ## 198 0.0575873399 2.052855e-03 1.862498e-03 1.528821e-02 4.902421e-03 ## 199 -0.0793128692 1.471287e-03 1.225470e-03 3.021010e-02 4.805799e-03 ## 200 0.0859659703 1.650029e-02 2.689987e-03 3.261799e-03 5.199372e-03 ## 201 -0.0669411018 -5.826575e-03 -1.506148e-03 -2.083865e-03 -2.624838e-03 ## 202 0.0014111781 2.651661e-03 2.512304e-02 1.960881e-03 5.650899e-03 ## 203 -0.4474588789 -1.002904e-03 5.771236e-02 -4.603904e-03 -3.966592e-03 ## 204 -0.1271436154 2.742472e-03 2.579109e-03 1.222930e-03 1.343826e-01 ## 205 0.0864409392 -6.421267e-02 -6.436043e-02 -6.416379e-02 -6.242336e-02 ## 206 0.1243156876 7.831292e-03 7.806819e-03 3.363287e-02 7.453436e-03 ## 207 -0.1847362688 6.561829e-03 6.755938e-03 5.749531e-02 1.920391e-03 ## 208 -0.0932978399 4.242700e-03 4.218592e-02 3.509214e-03 5.211666e-03 ## 209 -0.0024933586 4.295848e-03 3.582015e-02 3.835110e-03 7.126140e-03 ## 210 -0.3167313933 1.690724e-03 1.698006e-03 6.077838e-02 2.435018e-04 ## 211 -0.1123265716 7.723547e-03 5.176142e-02 8.631538e-03 4.708713e-03 ## 212 0.1269876929 1.033120e-02 2.732744e-03 3.808595e-03 4.864434e-03 ## 213 0.3121278640 -4.509419e-02 -4.543217e-02 -4.349786e-02 -3.987360e-02 ## 214 0.1959455631 4.366015e-03 1.727443e-02 5.440620e-03 8.549881e-03 ## 215 -0.0471733816 6.951487e-03 5.118209e-02 7.163267e-03 7.535401e-03 ## 216 -0.4304522595 8.436575e-02 7.398772e-03 6.762152e-03 -2.345916e-03 ## 217 -0.3527805048 4.029825e-03 6.657628e-02 2.169673e-03 6.545913e-04 ## 218 0.2911885895 -1.317274e-02 -1.337427e-02 -1.089328e-02 -9.900192e-03 ## 219 -0.0364364794 6.694750e-02 8.432362e-03 9.114545e-03 9.037317e-03 ## 220 -0.5883389632 9.720141e-02 7.400428e-03 6.305674e-03 -6.535765e-03 ## 221 -0.0704138236 8.012264e-03 7.894107e-03 8.193403e-03 1.841268e-01 ## 222 0.3193725815 4.852984e-02 1.003226e-02 1.302710e-02 1.600634e-02 ## 223 0.2171229626 1.103955e-02 1.071203e-02 6.103924e-02 1.521116e-02 ## hgt wgt hgt:wgt ## 1 -1.839190e-03 -1.790011e-03 1.389532e-05 ## 2 9.885098e-04 -2.377479e-03 9.303061e-06 ## 3 1.035244e-03 2.803606e-03 -1.608245e-05 ## 4 4.883509e-04 1.847202e-03 -1.001945e-05 ## 5 -9.320126e-04 -1.968394e-02 9.719343e-05 ## 6 9.077265e-04 3.892513e-03 -2.022645e-05 ## 7 3.391486e-04 1.610804e-03 -8.082354e-06 ## 8 -3.786088e-03 -1.330324e-02 7.466213e-05 ## 9 9.657141e-04 -6.058856e-05 -2.346532e-06 ## 10 7.916492e-04 1.846350e-03 -1.047630e-05 ## 11 4.002874e-04 1.430116e-03 -6.953553e-06 ## 12 7.140215e-04 4.482801e-03 -2.169506e-05 ## 13 -1.051408e-03 -3.453610e-03 1.944663e-05 ## 14 -5.177075e-04 -1.661756e-03 9.356349e-06 ## 15 4.404632e-04 -6.707243e-04 3.267699e-06 ## 16 -1.394320e-04 -4.880122e-04 2.679466e-06 ## 17 6.474481e-04 2.604471e-03 -1.340075e-05 ## 18 5.506830e-04 3.097835e-03 -1.526056e-05 ## 19 3.494419e-04 1.613877e-03 -8.391319e-06 ## 20 4.136425e-04 9.720525e-04 -5.639106e-06 ## 21 5.597618e-04 1.146061e-03 -6.632939e-06 ## 22 -8.467119e-04 -2.535326e-03 1.446626e-05 ## 23 5.091669e-04 8.961502e-04 -5.425561e-06 ## 24 2.397981e-04 8.808164e-04 -4.769228e-06 ## 25 3.556795e-04 1.424280e-03 -7.392609e-06 ## 26 5.559661e-05 -7.849560e-04 5.058486e-06 ## 27 -3.288173e-04 -6.110306e-04 3.816895e-06 ## 28 -2.011056e-03 -6.636789e-03 3.722996e-05 ## 29 -2.687933e-04 -7.379379e-04 4.190778e-06 ## 30 6.597567e-04 -1.201334e-03 3.748072e-06 ## 31 -6.298535e-04 -9.004975e-04 6.024807e-06 ## 32 3.985249e-05 9.561799e-04 -3.477617e-06 ## 33 -1.399723e-03 -1.000583e-03 1.231015e-05 ## 34 -9.988468e-06 -2.927674e-03 1.472505e-05 ## 35 2.265284e-04 8.059369e-04 -4.358269e-06 ## 36 2.088997e-04 -8.413796e-04 4.020492e-06 ## 37 1.177603e-04 -1.994820e-03 9.556883e-06 ## 38 -7.840950e-05 1.356594e-04 8.096808e-07 ## 39 -1.452807e-04 -1.853884e-04 1.187202e-06 ## 40 -1.953704e-04 -1.141769e-03 5.647074e-06 ## 41 4.435523e-05 1.596115e-04 -7.978551e-07 ## 42 1.685376e-04 -4.723391e-03 2.204398e-05 ## 43 2.716554e-06 -1.890219e-06 6.890237e-09 ## 44 -1.422959e-03 -4.143725e-03 2.323193e-05 ## 45 -1.058862e-04 -9.577555e-04 5.676680e-06 ## 46 -2.398132e-03 -6.131358e-03 3.573215e-05 ## 47 -1.015828e-04 -2.251173e-03 1.157204e-05 ## 48 -8.646389e-04 -1.498973e-03 9.148235e-06 ## 49 -3.215761e-03 -1.325805e-02 7.132016e-05 ## 50 8.414847e-05 -1.192507e-03 5.622315e-06 ## 51 -1.353953e-03 -2.374622e-03 1.775374e-05 ## 52 8.734106e-05 -4.934150e-03 2.293118e-05 ## 53 -2.229240e-04 -2.442728e-03 1.308402e-05 ## 54 -1.082491e-03 -3.037036e-03 1.967874e-05 ## 55 2.325737e-06 -6.762152e-04 3.312940e-06 ## 56 -2.418223e-04 -2.475643e-03 1.315300e-05 ## 57 -4.143487e-06 2.499598e-05 -1.625555e-07 ## 58 -4.363224e-05 -1.075985e-03 5.835192e-06 ## 59 -8.918594e-05 3.762440e-04 -9.127512e-07 ## 60 2.916254e-06 -9.703320e-05 3.585213e-07 ## 61 -8.952420e-04 -2.322686e-03 1.577534e-05 ## 62 -6.740786e-04 -3.623066e-03 1.849959e-05 ## 63 -5.695730e-05 -1.933721e-04 6.833537e-07 ## 64 -3.743138e-05 -9.778822e-04 5.213395e-06 ## 65 -3.594441e-03 -8.752983e-03 5.178472e-05 ## 66 -2.530220e-05 -1.255402e-03 6.413023e-06 ## 67 -2.705208e-04 -5.319411e-04 2.567364e-06 ## 68 1.043569e-05 -6.115551e-04 2.389427e-06 ## 69 3.037056e-05 -6.417080e-04 2.483484e-06 ## 70 -4.132345e-04 -2.401399e-03 1.340836e-05 ## 71 2.860871e-05 -5.167165e-04 1.739063e-06 ## 72 -2.258303e-04 -1.054393e-03 6.153124e-06 ## 73 -1.880679e-04 1.852608e-03 -8.818144e-06 ## 74 -7.074876e-04 -1.561124e-03 1.073596e-05 ## 75 -4.870484e-04 -1.317541e-02 6.298687e-05 ## 76 -6.796137e-04 -4.427300e-03 2.403196e-05 ## 77 -1.522258e-03 -3.774994e-03 2.371172e-05 ## 78 -5.471292e-04 -7.245038e-05 3.067103e-06 ## 79 -9.456802e-04 -4.721134e-03 2.604109e-05 ## 80 -6.693057e-04 -4.864560e-03 2.551639e-05 ## 81 1.064051e-04 1.759885e-04 -1.517005e-06 ## 82 4.959080e-04 -2.598996e-02 1.136725e-04 ## 83 -1.646557e-03 -5.895387e-03 3.384508e-05 ## 84 -1.086797e-03 -4.127418e-03 2.348755e-05 ## 85 1.230578e-04 3.509076e-04 -2.435227e-06 ## 86 -3.788191e-04 -5.458911e-03 2.431863e-05 ## 87 1.990171e-04 8.652712e-04 -5.180162e-06 ## 88 -1.821261e-04 -1.486554e-03 7.797844e-06 ## 89 -8.010847e-04 -9.677378e-03 4.778873e-05 ## 90 -1.448699e-03 -7.901499e-03 4.073302e-05 ## 91 -2.517746e-03 1.057314e-03 4.215545e-06 ## 92 2.551226e-04 6.626936e-04 -4.574886e-06 ## 93 -1.012790e-03 -3.518043e-03 2.051089e-05 ## 94 5.636885e-04 2.258005e-03 -1.426040e-05 ## 95 1.133617e-04 -5.101660e-04 1.731313e-06 ## 96 -2.143495e-04 -1.456154e-05 1.086514e-06 ## 97 3.203113e-04 2.591454e-04 -2.827783e-06 ## 98 2.714021e-04 1.638324e-03 -9.001136e-06 ## 99 -2.092537e-03 8.267623e-04 4.217711e-06 ## 100 1.582332e-04 1.789622e-03 -1.023219e-05 ## 101 3.686553e-04 -9.466831e-03 3.833117e-05 ## 102 -4.227996e-04 -2.731811e-04 3.155272e-06 ## 103 6.113337e-04 -1.007219e-03 5.418861e-07 ## 104 7.354426e-04 1.647233e-03 -1.162056e-05 ## 105 -1.206212e-03 1.042386e-03 2.652308e-07 ## 106 -9.853882e-04 -1.029242e-03 9.214359e-06 ## 107 -1.760687e-03 1.264744e-03 -3.025890e-07 ## 108 -9.521677e-04 -5.814403e-04 4.769345e-07 ## 109 3.757359e-04 3.745068e-03 -1.922376e-05 ## 110 -4.193571e-04 -1.432881e-03 8.319695e-06 ## 111 -4.896351e-04 -2.188556e-03 1.232215e-05 ## 112 -6.398442e-04 -6.274511e-04 5.582830e-06 ## 113 5.347542e-04 1.399294e-03 -9.518641e-06 ## 114 -3.868035e-04 -1.126680e-03 6.757458e-06 ## 115 2.886734e-04 6.738201e-04 -4.425114e-06 ## 116 1.026902e-04 3.215635e-03 -1.631969e-05 ## 117 -3.516825e-04 -1.220874e-03 7.019016e-06 ## 118 1.352561e-04 3.380964e-04 -2.170639e-06 ## 119 7.106117e-05 3.834745e-04 -2.075915e-06 ## 120 5.146115e-04 3.882232e-03 -2.032875e-05 ## 121 2.391138e-04 -1.305056e-03 1.344262e-06 ## 122 -4.657015e-04 -4.515276e-03 2.253990e-05 ## 123 -8.275184e-04 9.786764e-05 2.943044e-06 ## 124 -9.955660e-05 -1.111364e-03 5.491831e-06 ## 125 -1.212254e-03 2.630851e-03 -7.916104e-06 ## 126 2.533107e-04 5.065692e-04 -3.435988e-06 ## 127 5.707649e-04 6.356813e-04 -5.465480e-06 ## 128 -2.351121e-04 -8.923877e-04 5.126113e-06 ## 129 -6.432919e-04 -1.664396e-03 9.129728e-06 ## 130 -2.418433e-04 -2.178052e-03 1.071050e-05 ## 131 -2.437900e-04 -2.931912e-03 1.430538e-05 ## 132 -4.692637e-04 -2.511492e-04 2.804493e-06 ## 133 7.532897e-04 -1.600832e-02 6.741961e-05 ## 134 -6.589281e-04 -1.758259e-04 3.230984e-06 ## 135 5.064518e-03 9.591193e-03 -7.363051e-05 ## 136 -1.055851e-03 2.559743e-03 -7.876040e-06 ## 137 -4.429785e-04 3.233460e-04 2.115009e-07 ## 138 -1.059326e-04 -4.079979e-03 1.751692e-05 ## 139 -3.000385e-04 -2.752228e-04 2.316649e-06 ## 140 -2.425789e-04 -7.482223e-05 1.289727e-06 ## 141 3.510285e-03 7.798246e-03 -5.770248e-05 ## 142 -2.418102e-04 -8.684437e-04 4.800627e-06 ## 143 3.671228e-04 5.218044e-04 -3.883803e-06 ## 144 8.258265e-04 4.063244e-03 -2.473919e-05 ## 145 1.333797e-03 1.261163e-02 -6.730850e-05 ## 146 5.797013e-04 7.618688e-04 -5.915451e-06 ## 147 1.514522e-03 1.748951e-03 -1.529571e-05 ## 148 -4.230239e-04 5.383845e-04 -8.293325e-07 ## 149 6.704147e-05 3.797057e-03 -2.007958e-05 ## 150 5.185431e-04 2.475698e-04 -3.241917e-06 ## 151 -5.267093e-04 2.797423e-03 -1.220837e-05 ## 152 2.834813e-04 1.064711e-03 -5.945126e-06 ## 153 -1.269483e-04 -4.769732e-04 2.594917e-06 ## 154 1.438198e-04 8.490508e-05 -9.200474e-07 ## 155 -8.100004e-06 -2.556329e-05 1.391915e-07 ## 156 5.093397e-04 1.977506e-03 -1.103712e-05 ## 157 8.219034e-05 -4.927114e-05 -4.712173e-08 ## 158 6.810673e-06 7.913082e-06 -6.190054e-08 ## 159 9.784564e-04 7.775326e-03 -4.034260e-05 ## 160 -4.438859e-04 1.913932e-03 -7.339212e-06 ## 161 -1.772440e-04 2.634530e-03 -1.293714e-05 ## 162 -3.664596e-04 7.077530e-04 -2.119501e-06 ## 163 -1.132594e-04 1.186690e-03 -5.883363e-06 ## 164 -1.930979e-04 2.539212e-03 -1.247835e-05 ## 165 1.275423e-02 3.192814e-02 -2.130257e-04 ## 166 -1.905839e-04 6.980010e-04 -2.641131e-06 ## 167 1.157156e-04 1.470248e-04 -2.588624e-06 ## 168 5.106315e-04 1.908747e-04 -2.793956e-06 ## 169 7.109910e-04 3.889876e-03 -2.055291e-05 ## 170 1.015984e-03 1.359159e-03 -1.028812e-05 ## 171 -4.393062e-05 1.097842e-02 -5.292812e-05 ## 172 2.679460e-04 1.682160e-03 -8.425940e-06 ## 173 -2.173835e-04 1.389227e-03 -5.897037e-06 ## 174 1.853417e-04 5.885348e-04 -3.201669e-06 ## 175 1.366863e-04 1.929354e-04 -1.198884e-06 ## 176 3.639895e-03 1.728054e-02 -1.009949e-04 ## 177 -9.078355e-05 1.336349e-03 -6.102710e-06 ## 178 1.064416e-04 2.969717e-04 -2.637496e-06 ## 179 2.879511e-04 -2.982415e-04 5.688841e-07 ## 180 2.582356e-04 9.620222e-04 -4.864813e-06 ## 181 1.611714e-04 -5.353429e-04 2.260214e-06 ## 182 7.540996e-05 -1.386653e-04 6.243132e-07 ## 183 -4.427172e-05 8.298046e-04 -2.997981e-06 ## 184 2.960994e-04 -7.706919e-04 2.997738e-06 ## 185 5.120455e-04 1.018627e-02 -4.777362e-05 ## 186 2.730287e-03 4.687075e-03 -3.400372e-05 ## 187 3.964215e-05 -6.490287e-04 3.241376e-06 ## 188 5.208902e-04 6.482982e-03 -3.123253e-05 ## 189 -2.715073e-04 4.109324e-03 -1.699878e-05 ## 190 7.641955e-04 7.109810e-03 -3.798174e-05 ## 191 -4.654979e-04 2.112951e-03 -6.695036e-06 ## 192 1.323491e-03 9.770243e-03 -4.934055e-05 ## 193 1.745018e-03 7.813099e-03 -4.252136e-05 ## 194 9.882677e-04 2.124287e-03 -1.236349e-05 ## 195 -3.734086e-04 -1.718168e-03 1.060569e-05 ## 196 1.198338e-04 -3.665659e-03 1.737006e-05 ## 197 2.261074e-03 2.610864e-03 -2.086717e-05 ## 198 -1.363932e-04 -4.735449e-03 2.372084e-05 ## 199 1.060651e-03 -6.315818e-03 2.644677e-05 ## 200 -5.034598e-04 -3.329413e-03 1.890668e-05 ## 201 4.714585e-04 1.289619e-03 -8.578330e-06 ## 202 2.815626e-04 -4.943479e-03 2.349181e-05 ## 203 3.086666e-03 4.342903e-03 -3.205424e-05 ## 204 1.063851e-03 -2.633567e-03 9.899038e-06 ## 205 -1.418951e-04 -1.997947e-03 1.187337e-05 ## 206 -1.643859e-03 4.286136e-03 -9.897000e-06 ## 207 1.643787e-04 1.150571e-02 -5.089105e-05 ## 208 5.616044e-04 -4.694573e-04 2.406652e-06 ## 209 1.563272e-04 -3.944286e-03 2.015027e-05 ## 210 2.106572e-03 2.744930e-03 -1.959069e-05 ## 211 -2.220781e-04 8.904947e-03 -3.671241e-05 ## 212 -8.829739e-04 -2.517644e-03 1.649828e-05 ## 213 -1.651665e-03 -7.281428e-03 4.330963e-05 ## 214 -1.176929e-03 -6.101821e-03 3.511944e-05 ## 215 -7.372561e-05 1.532717e-03 -3.202884e-06 ## 216 1.280226e-03 2.038736e-02 -9.624929e-05 ## 217 1.886489e-03 7.476284e-03 -3.966297e-05 ## 218 -1.998752e-03 -3.699855e-03 2.735388e-05 ## 219 -2.990100e-04 2.417735e-03 -5.668995e-06 ## 220 1.873111e-03 2.784920e-02 -1.334001e-04 ## 221 -6.481672e-06 2.292112e-03 -6.433318e-06 ## 222 -2.380375e-03 -6.115464e-03 4.277606e-05 ## 223 -1.863021e-03 -3.300106e-03 2.806370e-05
Let’s use the simpler anscombe
data example
y_hat <- fit %>% fitted.values()
The residual is then calculated as
y_hat - anscombe$y1
## 1 2 3 4 5 6 ## -0.03900000 0.05081818 1.92127273 -1.30909091 0.17109091 0.04136364 ## 7 8 9 10 11 ## -1.23936364 0.74045455 -1.83881818 1.68072727 -0.17945455
If we introduce new values for the predictor x1
, we can generate predicted values from the model
new.x1 <- data.frame(x1 = 1:20) predict.lm(fit, newdata = new.x1)
## 1 2 3 4 5 6 7 ## 3.500182 4.000273 4.500364 5.000455 5.500545 6.000636 6.500727 ## 8 9 10 11 12 13 14 ## 7.000818 7.500909 8.001000 8.501091 9.001182 9.501273 10.001364 ## 15 16 17 18 19 20 ## 10.501455 11.001545 11.501636 12.001727 12.501818 13.001909
pred <- predict.lm(fit, newdata = new.x1) lm(pred ~ new.x1$x1)$coefficients
## (Intercept) new.x1$x1 ## 3.0000909 0.5000909
fit$coefficients
## (Intercept) x1 ## 3.0000909 0.5000909
predict(fit, interval = "prediction")
## fit lwr upr ## 1 8.001000 5.067072 10.934928 ## 2 7.000818 4.066890 9.934747 ## 3 9.501273 6.390801 12.611745 ## 4 7.500909 4.579129 10.422689 ## 5 8.501091 5.531014 11.471168 ## 6 10.001364 6.789620 13.213107 ## 7 6.000636 2.971271 9.030002 ## 8 5.000455 1.788711 8.212198 ## 9 9.001182 5.971816 12.030547 ## 10 6.500727 3.530650 9.470804 ## 11 5.500545 2.390073 8.611017
Do for all \(j\in\{1,\dots,k\}\) training sets
Small difference suggests good predictive accuracy
anscombe
dataDAAG::CVlm(anscombe, fit, plotit=T, printit=F)
boys
dataDAAG::CVlm(na.omit(boys), boys.fit3, plotit=T, printit=F)
na.omit(boys) %$% lm(age ~ reg + hgt * wgt) %>% nobs()
## [1] 223
If we would not have used na.omit()
boys %$% lm(age ~ reg + hgt * wgt) %>% nobs()
## [1] 724
R
lm()
: linear modelingglm()
: generalized linear modelinggamlss::gamlss
: generalized additive models (for location, scale and shape)lme4::lmer
: linear mixed effect modelslme4::glmer
: generalized linear mixed effect modelslme4::nlmer
: non-linear mixed effect models