Multiple Regression
Dieter 2022-08-26
- Data used
- Read in some data
- Revisit the simple linear model
- Intro to the multiple linear regression
- Model comparison
- Interactions
- Step
Data used
body.csvvik_table_9_2.csv
Read in some data
library(tidyverse)
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## ✔ tibble 3.1.8 ✔ dplyr 1.0.10
## ✔ tidyr 1.2.1 ✔ stringr 1.4.1
## ✔ readr 2.1.3 ✔ forcats 0.5.2
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body_data <-read_csv('data/body.csv')
## Rows: 507 Columns: 25
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## dbl (25): Biacromial, Biiliac, Bitrochanteric, ChestDepth, ChestDia, ElbowDi...
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
Revisit the simple linear model
This is q quick recap of the simple linear regression model.
model <- lm(Weight ~ Waist, data=body_data)
summary(model)
##
## Call:
## lm(formula = Weight ~ Waist, data = body_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.3211 -3.5995 -0.0936 3.6932 19.0536
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -15.18382 1.79291 -8.469 2.72e-16 ***
## Waist 1.09550 0.02306 47.514 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.712 on 505 degrees of freedom
## Multiple R-squared: 0.8172, Adjusted R-squared: 0.8168
## F-statistic: 2258 on 1 and 505 DF, p-value: < 2.2e-16
The anova table shows more details about the components calculating the F-value
anova(model)
## Analysis of Variance Table
##
## Response: Weight
## Df Sum Sq Mean Sq F value Pr(>F)
## Waist 1 73649 73649 2257.6 < 2.2e-16 ***
## Residuals 505 16475 33
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Manual calculation of the F value
As we mentioned before, R automatically calculates the F-value. But we can also do this manually. You an inspect this code if you want to understand the calculation of the F value in more detail.
predicted <- predict(model)
mn_predicted <-mean(predicted)
resids <- model$residuals
captured_variance <- sum((predicted - mn_predicted)^2)
uncaptured_variance <- sum((resids)^2)
n_predictors <- 1
manual_f <- (captured_variance / n_predictors)/ (uncaptured_variance/ model$df.residual)
captured_variance
## [1] 73648.74
uncaptured_variance
## [1] 16474.61
manual_f
## [1] 2257.572
Intro to the multiple linear regression
A first example
Here I use data provided by Vik 2013 (Table 9.2 in Vik, Peter. Regression, ANOVA, and the General Linear Model, 2013.). This allows double checking the calculating provided in this book.
vik_data <-read_csv('data/vik_table_9_2.csv')
## Rows: 12 Columns: 4
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## dbl (4): Person, Y, X1, X2
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
head(vik_data)
## # A tibble: 6 × 4
## Person Y X1 X2
## <dbl> <dbl> <dbl> <dbl>
## 1 1 2 8 1
## 2 2 3 9 2
## 3 3 3 9 2
## 4 4 4 10 3
## 5 5 7 6 8
## 6 6 5 7 9
model <- lm(Y ~ X1 + X2, data=vik_data)
summary(model)
##
## Call:
## lm(formula = Y ~ X1 + X2, data = vik_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.45564 -0.24631 0.03248 0.54747 1.52060
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.2014 1.5185 6.060 0.000188 ***
## X1 -0.7085 0.1481 -4.783 0.000997 ***
## X2 0.1209 0.1486 0.813 0.436963
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.271 on 9 degrees of freedom
## Multiple R-squared: 0.8182, Adjusted R-squared: 0.7777
## F-statistic: 20.25 on 2 and 9 DF, p-value: 0.0004663
Manual calculation of the F value
Just as before, we can calculate the F value manually to get a better understanding of the calculation of F.
predicted <- predict(model)
mn_predicted <-mean(predicted)
resids <- model$residuals
captured_variance <- sum((predicted - mn_predicted)^2)
uncaptured_variance <- sum((resids)^2)
n_predictors <- 2
manual_F <- (captured_variance / n_predictors)/ (uncaptured_variance/ model$df.residual)
captured_variance
## [1] 65.45267
uncaptured_variance
## [1] 14.54733
manual_f
## [1] 2257.572
Visualizing the fitted model
# library(plotly)
# library(pracma)
# coeffs <- model$coefficient
# xi <- seq(min(vik_data$X1), max(vik_data$X1), length.out=100)
# yi <- seq(min(vik_data$X2), max(vik_data$X2), length.out=100)
# grid <- meshgrid(xi,yi)
# xi <- grid[[1]]
# yi <- grid[[2]]
#
#
# zi <- coeffs[1] + coeffs[2] * xi + coeffs[3] * yi
#
# my_plot <- plot_ly(vik_data, x = ~X1, y = ~X2, z = ~Y, type = "scatter3d", mode = "markers")
# my_plot <- add_trace(p = my_plot, z = zi, x = xi, y = yi, type = "surface")
# my_plot
Manually running the omnibus test
I fit the full model and the base model (which is usually only done implicitly). We fit the following two models:
\[y_i = \bar{y}\] \[y_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i}\]Next, we explicitly compare their performances using the anova
function. This results in the same F-value as the one calculated by R
when fitting the full model.
vik_data$mean_y <- mean(vik_data$Y)
model_base <- lm(Y ~ mean_y, data=vik_data)
model_full <- lm(Y ~ X1 + X2, data=vik_data)
Let’s look at the full model’s F value.
summary(model_full)
##
## Call:
## lm(formula = Y ~ X1 + X2, data = vik_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.45564 -0.24631 0.03248 0.54747 1.52060
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.2014 1.5185 6.060 0.000188 ***
## X1 -0.7085 0.1481 -4.783 0.000997 ***
## X2 0.1209 0.1486 0.813 0.436963
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.271 on 9 degrees of freedom
## Multiple R-squared: 0.8182, Adjusted R-squared: 0.7777
## F-statistic: 20.25 on 2 and 9 DF, p-value: 0.0004663
Let’s now explicitly compare the base and the full model. This results in the same F-value! So, we could run the omnibus test by comparing the base model, $y_i = \bar{x}$, with the full model. But this is done implicitly by R when fitting the full model.
anova(model_base, model_full)
## Analysis of Variance Table
##
## Model 1: Y ~ mean_y
## Model 2: Y ~ X1 + X2
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 11 80.000
## 2 9 14.547 2 65.453 20.247 0.0004663 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Manually calculating the F ratio for the omnibus test
full_model_predictions <- predict(model_full)
mean_full_model_predictions <-mean(full_model_predictions)
base_model_predictions <- predict(model_base)
captured_variance <- sum((full_model_predictions - mean_full_model_predictions)^2)
uncaptured_variance <- sum((base_model_predictions- vik_data$Y)^2)
test <- sum((model_full$residuals)^2)
test
## [1] 14.54733
captured_variance
## [1] 65.45267
uncaptured_variance
## [1] 80
manual_f <- captured_variance/uncaptured_variance
manual_f
## [1] 0.8181584
Model comparison
Here, we fit two nested models and compare their peformance using the
anova function.
A simple and a complex model
state_data <- as.data.frame(state.x77)
colnames(state_data) <- make.names(colnames(state_data))
head(state_data)
## Population Income Illiteracy Life.Exp Murder HS.Grad Frost Area
## Alabama 3615 3624 2.1 69.05 15.1 41.3 20 50708
## Alaska 365 6315 1.5 69.31 11.3 66.7 152 566432
## Arizona 2212 4530 1.8 70.55 7.8 58.1 15 113417
## Arkansas 2110 3378 1.9 70.66 10.1 39.9 65 51945
## California 21198 5114 1.1 71.71 10.3 62.6 20 156361
## Colorado 2541 4884 0.7 72.06 6.8 63.9 166 103766
Let’s fit two (nested) models,
simple_model <- lm(state_data$Murder ~ state_data$Illiteracy)
complex_model <- lm(state_data$Murder ~ state_data$Illiteracy + state_data$Income + state_data$Population)
Let’s look at their summaries:
summary(simple_model)
##
## Call:
## lm(formula = state_data$Murder ~ state_data$Illiteracy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.5315 -2.0602 -0.2503 1.6916 6.9745
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.3968 0.8184 2.928 0.0052 **
## state_data$Illiteracy 4.2575 0.6217 6.848 1.26e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.653 on 48 degrees of freedom
## Multiple R-squared: 0.4942, Adjusted R-squared: 0.4836
## F-statistic: 46.89 on 1 and 48 DF, p-value: 1.258e-08
summary(complex_model)
##
## Call:
## lm(formula = state_data$Murder ~ state_data$Illiteracy + state_data$Income +
## state_data$Population)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.7846 -1.6768 -0.0839 1.4783 7.6417
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.3402721 3.3694210 0.398 0.6926
## state_data$Illiteracy 4.1109188 0.6706786 6.129 1.85e-07 ***
## state_data$Income 0.0000644 0.0006762 0.095 0.9245
## state_data$Population 0.0002219 0.0000842 2.635 0.0114 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.507 on 46 degrees of freedom
## Multiple R-squared: 0.5669, Adjusted R-squared: 0.5387
## F-statistic: 20.07 on 3 and 46 DF, p-value: 1.84e-08
anova(simple_model, complex_model)
## Analysis of Variance Table
##
## Model 1: state_data$Murder ~ state_data$Illiteracy
## Model 2: state_data$Murder ~ state_data$Illiteracy + state_data$Income +
## state_data$Population
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 48 337.76
## 2 46 289.19 2 48.574 3.8633 0.02812 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
What if the models only differ by one variable
In this case, the p-value associated with the newly introduced variable is equal to the p value of the anova comparison. In fact the $F$ value is equal to $t^2$ value.
simple_model <- lm(state_data$Murder ~ state_data$Illiteracy)
complex_model <- lm(state_data$Murder ~ state_data$Illiteracy + state_data$Income)
summary(complex_model)
##
## Call:
## lm(formula = state_data$Murder ~ state_data$Illiteracy + state_data$Income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.6343 -1.9289 -0.0171 1.6779 6.7349
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.4409926 3.5034602 -0.126 0.900
## state_data$Illiteracy 4.5099882 0.6934465 6.504 4.63e-08 ***
## state_data$Income 0.0005731 0.0006879 0.833 0.409
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.661 on 47 degrees of freedom
## Multiple R-squared: 0.5015, Adjusted R-squared: 0.4803
## F-statistic: 23.64 on 2 and 47 DF, p-value: 7.841e-08
anova(simple_model, complex_model)
## Analysis of Variance Table
##
## Model 1: state_data$Murder ~ state_data$Illiteracy
## Model 2: state_data$Murder ~ state_data$Illiteracy + state_data$Income
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 48 337.76
## 2 47 332.85 1 4.9163 0.6942 0.4089
Interactions
Model without interaction
library(gapminder)
gap <- gapminder
model <- lm(lifeExp ~ gdpPercap + year, data = gap)
summary(model)
##
## Call:
## lm(formula = lifeExp ~ gdpPercap + year, data = gap)
##
## Residuals:
## Min 1Q Median 3Q Max
## -67.262 -6.954 1.219 7.759 19.553
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.184e+02 2.762e+01 -15.15 <2e-16 ***
## gdpPercap 6.697e-04 2.447e-05 27.37 <2e-16 ***
## year 2.390e-01 1.397e-02 17.11 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9.694 on 1701 degrees of freedom
## Multiple R-squared: 0.4375, Adjusted R-squared: 0.4368
## F-statistic: 661.4 on 2 and 1701 DF, p-value: < 2.2e-16
Model with interaction
model <- lm(lifeExp ~ gdpPercap * year, data = gap)
summary(model)
##
## Call:
## lm(formula = lifeExp ~ gdpPercap * year, data = gap)
##
## Residuals:
## Min 1Q Median 3Q Max
## -54.234 -7.314 1.002 7.951 19.780
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.532e+02 3.267e+01 -10.811 < 2e-16 ***
## gdpPercap -8.754e-03 2.547e-03 -3.437 0.000602 ***
## year 2.060e-01 1.653e-02 12.463 < 2e-16 ***
## gdpPercap:year 4.754e-06 1.285e-06 3.700 0.000222 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9.658 on 1700 degrees of freedom
## Multiple R-squared: 0.442, Adjusted R-squared: 0.441
## F-statistic: 448.8 on 3 and 1700 DF, p-value: < 2.2e-16
This gives the same result:
gap$multi <- gap$gdpPercap * gap$year
model <- lm(lifeExp ~ gdpPercap + year + multi, data = gap)
summary(model)
##
## Call:
## lm(formula = lifeExp ~ gdpPercap + year + multi, data = gap)
##
## Residuals:
## Min 1Q Median 3Q Max
## -54.234 -7.314 1.002 7.951 19.780
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.532e+02 3.267e+01 -10.811 < 2e-16 ***
## gdpPercap -8.754e-03 2.547e-03 -3.437 0.000602 ***
## year 2.060e-01 1.653e-02 12.463 < 2e-16 ***
## multi 4.754e-06 1.285e-06 3.700 0.000222 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9.658 on 1700 degrees of freedom
## Multiple R-squared: 0.442, Adjusted R-squared: 0.441
## F-statistic: 448.8 on 3 and 1700 DF, p-value: < 2.2e-16
range(gap$gdpPercap)
## [1] 241.1659 113523.1329
range(gap$year)
## [1] 1952 2007
See this link for a plot of the model: https://www.wolframalpha.com/input?i=plot+-3.532e%2B02+%2B+x+*+-8.754e-03+%2B+y+*+2.060e-01+%2B+4.754e-06+*+%28x+*+y%29+with+x++%3D+241+to+113523+and+y+%3D+1952+to+2007
Using zeroed data
gap <- mutate(gap, year = scale(year))
gap <- mutate(gap, gdpPercap = scale(gdpPercap))
model <- lm(lifeExp ~ gdpPercap * year, data = gap)
range(gap$gdpPercap)
## [1] -0.7075012 10.7845089
range(gap$year)
## [1] -1.592787 1.592787
summary(model)
##
## Call:
## lm(formula = lifeExp ~ gdpPercap * year, data = gap)
##
## Residuals:
## Min 1Q Median 3Q Max
## -54.234 -7.314 1.002 7.951 19.780
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 59.2906 0.2392 247.90 < 2e-16 ***
## gdpPercap 6.4680 0.2430 26.61 < 2e-16 ***
## year 4.1489 0.2404 17.26 < 2e-16 ***
## gdpPercap:year 0.8091 0.2187 3.70 0.000222 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9.658 on 1700 degrees of freedom
## Multiple R-squared: 0.442, Adjusted R-squared: 0.441
## F-statistic: 448.8 on 3 and 1700 DF, p-value: < 2.2e-16
The fitted model becomes,
\(z = 59.29 + 6.46 x + 4.14 y + 0.80 + x y\) with $x$ the gdpPercap and $y$ the year.
See this link for a graph: https://www.wolframalpha.com/input?i=plot+59.2906+%2B+6.4680+*+x+%2B+4.1489+*+y+%2B+0.8091+%2B+x+*+y+with+x+from+-+1+to+11+and+y+from+-2+to+2
Step
Using the Murder data
library(MASS)
##
## Attaching package: 'MASS'
## The following object is masked from 'package:dplyr':
##
## select
full=lm(Murder~.,data=state_data)
stepAIC(full, direction="backward", trace = TRUE)
## Start: AIC=63.01
## Murder ~ Population + Income + Illiteracy + Life.Exp + HS.Grad +
## Frost + Area
##
## Df Sum of Sq RSS AIC
## - Income 1 0.236 128.27 61.105
## - HS.Grad 1 0.973 129.01 61.392
## <none> 128.03 63.013
## - Area 1 7.514 135.55 63.865
## - Illiteracy 1 8.299 136.33 64.154
## - Frost 1 9.260 137.29 64.505
## - Population 1 25.719 153.75 70.166
## - Life.Exp 1 127.175 255.21 95.503
##
## Step: AIC=61.11
## Murder ~ Population + Illiteracy + Life.Exp + HS.Grad + Frost +
## Area
##
## Df Sum of Sq RSS AIC
## - HS.Grad 1 0.763 129.03 59.402
## <none> 128.27 61.105
## - Area 1 7.310 135.58 61.877
## - Illiteracy 1 8.715 136.98 62.392
## - Frost 1 9.345 137.61 62.621
## - Population 1 27.142 155.41 68.702
## - Life.Exp 1 127.500 255.77 93.613
##
## Step: AIC=59.4
## Murder ~ Population + Illiteracy + Life.Exp + Frost + Area
##
## Df Sum of Sq RSS AIC
## <none> 129.03 59.402
## - Illiteracy 1 8.723 137.75 60.672
## - Frost 1 11.030 140.06 61.503
## - Area 1 15.937 144.97 63.225
## - Population 1 26.415 155.45 66.714
## - Life.Exp 1 140.391 269.42 94.213
##
## Call:
## lm(formula = Murder ~ Population + Illiteracy + Life.Exp + Frost +
## Area, data = state_data)
##
## Coefficients:
## (Intercept) Population Illiteracy Life.Exp Frost Area
## 1.202e+02 1.780e-04 1.173e+00 -1.608e+00 -1.373e-02 6.804e-06
Using the body data
library(MASS)
full=lm(Weight~.,data=body_data)
stepAIC(full, direction="backward", trace = TRUE)
## Start: AIC=775.52
## Weight ~ Biacromial + Biiliac + Bitrochanteric + ChestDepth +
## ChestDia + ElbowDia + WristDia + KneeDia + AnkleDia + Shoulder +
## Chest + Waist + Navel + Hip + Thigh + Bicep + Forearm + Knee +
## Calf + Ankle + Wrist + Age + Height + Gender
##
## Df Sum of Sq RSS AIC
## - Ankle 1 0.06 2120.8 773.54
## - Navel 1 0.49 2121.2 773.64
## - Biacromial 1 0.71 2121.5 773.69
## - AnkleDia 1 2.08 2122.8 774.02
## - WristDia 1 3.42 2124.2 774.34
## - Bitrochanteric 1 3.83 2124.6 774.44
## - Wrist 1 5.98 2126.7 774.95
## - ElbowDia 1 6.57 2127.3 775.09
## - Bicep 1 8.17 2128.9 775.47
## <none> 2120.7 775.52
## - Biiliac 1 13.13 2133.9 776.65
## - ChestDia 1 15.71 2136.5 777.27
## - Shoulder 1 27.81 2148.6 780.13
## - Knee 1 30.39 2151.1 780.74
## - Gender 1 36.02 2156.8 782.06
## - KneeDia 1 43.48 2164.2 783.81
## - Chest 1 47.62 2168.4 784.78
## - Forearm 1 51.49 2172.2 785.69
## - ChestDepth 1 77.87 2198.6 791.81
## - Thigh 1 78.65 2199.4 791.99
## - Age 1 88.97 2209.7 794.36
## - Hip 1 108.21 2229.0 798.75
## - Calf 1 118.89 2239.6 801.18
## - Waist 1 797.97 2918.7 935.45
## - Height 1 1173.76 3294.5 996.85
##
## Step: AIC=773.54
## Weight ~ Biacromial + Biiliac + Bitrochanteric + ChestDepth +
## ChestDia + ElbowDia + WristDia + KneeDia + AnkleDia + Shoulder +
## Chest + Waist + Navel + Hip + Thigh + Bicep + Forearm + Knee +
## Calf + Wrist + Age + Height + Gender
##
## Df Sum of Sq RSS AIC
## - Navel 1 0.46 2121.3 771.65
## - Biacromial 1 0.69 2121.5 771.70
## - AnkleDia 1 2.46 2123.3 772.13
## - WristDia 1 3.37 2124.2 772.34
## - Bitrochanteric 1 3.80 2124.6 772.45
## - Wrist 1 6.08 2126.9 772.99
## - ElbowDia 1 6.52 2127.3 773.09
## - Bicep 1 8.15 2129.0 773.48
## <none> 2120.8 773.54
## - Biiliac 1 13.10 2133.9 774.66
## - ChestDia 1 15.71 2136.5 775.28
## - Shoulder 1 27.86 2148.7 778.15
## - Knee 1 32.75 2153.6 779.31
## - Gender 1 36.02 2156.8 780.08
## - KneeDia 1 43.87 2164.7 781.92
## - Chest 1 47.59 2168.4 782.79
## - Forearm 1 51.43 2172.2 783.69
## - ChestDepth 1 77.84 2198.6 789.81
## - Thigh 1 78.76 2199.6 790.02
## - Age 1 89.71 2210.5 792.54
## - Hip 1 108.59 2229.4 796.86
## - Calf 1 137.71 2258.5 803.44
## - Waist 1 798.17 2919.0 933.49
## - Height 1 1174.23 3295.0 994.93
##
## Step: AIC=771.65
## Weight ~ Biacromial + Biiliac + Bitrochanteric + ChestDepth +
## ChestDia + ElbowDia + WristDia + KneeDia + AnkleDia + Shoulder +
## Chest + Waist + Hip + Thigh + Bicep + Forearm + Knee + Calf +
## Wrist + Age + Height + Gender
##
## Df Sum of Sq RSS AIC
## - Biacromial 1 0.66 2121.9 769.81
## - AnkleDia 1 2.22 2123.5 770.18
## - Bitrochanteric 1 3.57 2124.8 770.50
## - WristDia 1 3.64 2124.9 770.52
## - Wrist 1 6.05 2127.3 771.09
## - ElbowDia 1 6.26 2127.5 771.14
## - Bicep 1 7.75 2129.0 771.50
## <none> 2121.3 771.65
## - Biiliac 1 12.67 2133.9 772.67
## - ChestDia 1 16.22 2137.5 773.51
## - Shoulder 1 28.62 2149.9 776.44
## - Knee 1 32.29 2153.6 777.31
## - Gender 1 36.57 2157.8 778.31
## - KneeDia 1 45.28 2166.6 780.36
## - Chest 1 47.13 2168.4 780.79
## - Forearm 1 52.51 2173.8 782.05
## - ChestDepth 1 77.39 2198.7 787.82
## - Thigh 1 81.25 2202.5 788.71
## - Age 1 96.82 2218.1 792.28
## - Hip 1 121.11 2242.4 797.80
## - Calf 1 140.97 2262.2 802.27
## - Waist 1 896.91 3018.2 948.44
## - Height 1 1174.61 3295.9 993.06
##
## Step: AIC=769.81
## Weight ~ Biiliac + Bitrochanteric + ChestDepth + ChestDia + ElbowDia +
## WristDia + KneeDia + AnkleDia + Shoulder + Chest + Waist +
## Hip + Thigh + Bicep + Forearm + Knee + Calf + Wrist + Age +
## Height + Gender
##
## Df Sum of Sq RSS AIC
## - AnkleDia 1 2.48 2124.4 768.40
## - WristDia 1 3.67 2125.6 768.68
## - Bitrochanteric 1 4.29 2126.2 768.83
## - ElbowDia 1 6.13 2128.1 769.27
## - Wrist 1 6.92 2128.9 769.46
## - Bicep 1 8.26 2130.2 769.78
## <none> 2121.9 769.81
## - Biiliac 1 12.23 2134.2 770.72
## - ChestDia 1 15.56 2137.5 771.51
## - Shoulder 1 28.16 2150.1 774.49
## - Knee 1 32.95 2154.9 775.62
## - Gender 1 40.48 2162.4 777.39
## - KneeDia 1 44.74 2166.7 778.39
## - Chest 1 48.22 2170.2 779.20
## - Forearm 1 52.71 2174.6 780.25
## - ChestDepth 1 77.31 2199.2 785.95
## - Thigh 1 81.31 2203.2 786.87
## - Age 1 96.41 2218.3 790.33
## - Hip 1 123.57 2245.5 796.50
## - Calf 1 140.68 2262.6 800.35
## - Waist 1 897.14 3019.1 946.59
## - Height 1 1219.78 3341.7 998.06
##
## Step: AIC=768.4
## Weight ~ Biiliac + Bitrochanteric + ChestDepth + ChestDia + ElbowDia +
## WristDia + KneeDia + Shoulder + Chest + Waist + Hip + Thigh +
## Bicep + Forearm + Knee + Calf + Wrist + Age + Height + Gender
##
## Df Sum of Sq RSS AIC
## - Bitrochanteric 1 4.47 2128.9 767.46
## - WristDia 1 4.54 2129.0 767.48
## - Wrist 1 6.39 2130.8 767.92
## - Bicep 1 7.91 2132.3 768.28
## <none> 2124.4 768.40
## - ElbowDia 1 9.46 2133.9 768.65
## - Biiliac 1 13.47 2137.9 769.60
## - ChestDia 1 15.43 2139.8 770.07
## - Shoulder 1 26.32 2150.7 772.64
## - Knee 1 31.67 2156.1 773.90
## - Gender 1 38.79 2163.2 775.57
## - Chest 1 51.66 2176.1 778.58
## - Forearm 1 51.96 2176.4 778.65
## - KneeDia 1 53.53 2177.9 779.02
## - ChestDepth 1 77.05 2201.5 784.46
## - Thigh 1 82.29 2206.7 785.67
## - Age 1 94.35 2218.8 788.43
## - Hip 1 122.97 2247.4 794.93
## - Calf 1 143.48 2267.9 799.53
## - Waist 1 896.31 3020.7 944.86
## - Height 1 1255.96 3380.4 1001.90
##
## Step: AIC=767.46
## Weight ~ Biiliac + ChestDepth + ChestDia + ElbowDia + WristDia +
## KneeDia + Shoulder + Chest + Waist + Hip + Thigh + Bicep +
## Forearm + Knee + Calf + Wrist + Age + Height + Gender
##
## Df Sum of Sq RSS AIC
## - WristDia 1 4.90 2133.8 766.63
## - Wrist 1 6.96 2135.8 767.12
## - Bicep 1 7.34 2136.2 767.21
## - ElbowDia 1 7.67 2136.6 767.29
## <none> 2128.9 767.46
## - Biiliac 1 9.65 2138.5 767.76
## - ChestDia 1 12.91 2141.8 768.53
## - Shoulder 1 27.13 2156.0 771.88
## - Knee 1 33.25 2162.1 773.32
## - Gender 1 39.54 2168.4 774.79
## - KneeDia 1 51.57 2180.5 777.60
## - Forearm 1 54.44 2183.3 778.27
## - Chest 1 57.69 2186.6 779.02
## - ChestDepth 1 77.06 2205.9 783.49
## - Thigh 1 87.77 2216.7 785.95
## - Age 1 99.38 2228.3 788.60
## - Hip 1 127.01 2255.9 794.84
## - Calf 1 139.66 2268.5 797.68
## - Waist 1 911.72 3040.6 946.19
## - Height 1 1254.46 3383.3 1000.34
##
## Step: AIC=766.63
## Weight ~ Biiliac + ChestDepth + ChestDia + ElbowDia + KneeDia +
## Shoulder + Chest + Waist + Hip + Thigh + Bicep + Forearm +
## Knee + Calf + Wrist + Age + Height + Gender
##
## Df Sum of Sq RSS AIC
## - Wrist 1 3.97 2137.7 765.57
## - Bicep 1 7.13 2140.9 766.32
## <none> 2133.8 766.63
## - Biiliac 1 9.23 2143.0 766.82
## - ElbowDia 1 12.10 2145.9 767.49
## - ChestDia 1 13.55 2147.3 767.84
## - Shoulder 1 26.56 2160.3 770.90
## - Knee 1 32.84 2166.6 772.37
## - Gender 1 39.48 2173.3 773.92
## - Forearm 1 54.13 2187.9 777.33
## - KneeDia 1 56.85 2190.6 777.96
## - Chest 1 60.08 2193.9 778.71
## - ChestDepth 1 73.88 2207.7 781.89
## - Thigh 1 85.73 2219.5 784.60
## - Age 1 97.74 2231.5 787.34
## - Hip 1 127.75 2261.5 794.11
## - Calf 1 142.03 2275.8 797.30
## - Waist 1 917.62 3051.4 945.99
## - Height 1 1271.05 3404.8 1001.55
##
## Step: AIC=765.57
## Weight ~ Biiliac + ChestDepth + ChestDia + ElbowDia + KneeDia +
## Shoulder + Chest + Waist + Hip + Thigh + Bicep + Forearm +
## Knee + Calf + Age + Height + Gender
##
## Df Sum of Sq RSS AIC
## - Bicep 1 6.68 2144.4 765.15
## <none> 2137.7 765.57
## - ElbowDia 1 11.05 2148.8 766.18
## - Biiliac 1 11.41 2149.2 766.27
## - ChestDia 1 14.04 2151.8 766.89
## - Shoulder 1 26.62 2164.4 769.85
## - Knee 1 31.10 2168.8 770.89
## - Gender 1 39.75 2177.5 772.91
## - Forearm 1 50.51 2188.3 775.41
## - KneeDia 1 53.43 2191.2 776.09
## - Chest 1 58.40 2196.1 777.24
## - ChestDepth 1 73.15 2210.9 780.63
## - Thigh 1 99.26 2237.0 786.58
## - Age 1 101.33 2239.1 787.05
## - Hip 1 124.78 2262.5 792.33
## - Calf 1 138.06 2275.8 795.30
## - Waist 1 926.25 3064.0 946.08
## - Height 1 1279.67 3417.4 1001.42
##
## Step: AIC=765.15
## Weight ~ Biiliac + ChestDepth + ChestDia + ElbowDia + KneeDia +
## Shoulder + Chest + Waist + Hip + Thigh + Forearm + Knee +
## Calf + Age + Height + Gender
##
## Df Sum of Sq RSS AIC
## <none> 2144.4 765.15
## - ElbowDia 1 11.03 2155.5 765.75
## - ChestDia 1 11.46 2155.9 765.86
## - Biiliac 1 12.19 2156.6 766.03
## - Knee 1 27.80 2172.2 769.68
## - Shoulder 1 32.32 2176.7 770.74
## - Gender 1 37.06 2181.5 771.84
## - KneeDia 1 54.73 2199.2 775.93
## - ChestDepth 1 71.64 2216.1 779.81
## - Chest 1 74.65 2219.1 780.50
## - Age 1 98.29 2242.7 785.88
## - Forearm 1 119.40 2263.8 790.63
## - Hip 1 121.76 2266.2 791.15
## - Thigh 1 123.73 2268.2 791.59
## - Calf 1 133.88 2278.3 793.86
## - Waist 1 929.27 3073.7 945.68
## - Height 1 1274.33 3418.8 999.62
##
## Call:
## lm(formula = Weight ~ Biiliac + ChestDepth + ChestDia + ElbowDia +
## KneeDia + Shoulder + Chest + Waist + Hip + Thigh + Forearm +
## Knee + Calf + Age + Height + Gender, data = body_data)
##
## Coefficients:
## (Intercept) Biiliac ChestDepth ChestDia ElbowDia KneeDia
## -121.11380 0.09451 0.28004 0.12415 0.26400 0.45002
## Shoulder Chest Waist Hip Thigh Forearm
## 0.07936 0.14703 0.37465 0.20624 0.26130 0.52963
## Knee Calf Age Height Gender
## 0.18622 0.34008 -0.05606 0.29486 -1.42332