Lab Worksheet 6
In 1898, Hermon Bumpus, an American biologist working at Brown University, collected data on one of the first examples of natural selection directly observed in nature. Immediately following a bad winter storm, he collected 136 English house sparrows, Passer domesticus, and brought them indoors. Of these birds, 64 had died during the storm, but 72 recovered and survived. By comparing measurements of physical traits, Bumpus demonstrated physical differences between the dead and living birds. He interpreted this finding as evidence for natural selection as a result of this storm:
bumpus <- read_csv("http://wilkelab.org/classes/SDS348/data_sets/bumpus_full.csv")## Parsed with column specification:
## cols(
## Sex = col_character(),
## Age = col_character(),
## Survival = col_character(),
## Length = col_integer(),
## Wingspread = col_integer(),
## Weight = col_double(),
## Skull_Length = col_double(),
## Humerus_Length = col_double(),
## Femur_Length = col_double(),
## Tarsus_Length = col_double(),
## Sternum_Length = col_double(),
## Skull_Width = col_double()
## )bumpus$Survival <- factor(bumpus$Survival)
head(bumpus)## # A tibble: 6 x 12
## Sex Age Survival Length Wingspread Weight Skull_Length Humerus_Length
## <chr> <chr> <fct> <int> <int> <dbl> <dbl> <dbl>
## 1 Male Adult Alive 154 241 24.5 31.2 17.4
## 2 Male Adult Alive 160 252 26.9 30.8 18.7
## 3 Male Adult Alive 155 243 26.9 30.6 18.6
## 4 Male Adult Alive 154 245 24.3 31.7 18.8
## 5 Male Adult Alive 156 247 24.1 31.5 18.2
## 6 Male Adult Alive 161 253 26.5 31.8 19.8
## # ... with 4 more variables: Femur_Length <dbl>, Tarsus_Length <dbl>,
## # Sternum_Length <dbl>, Skull_Width <dbl>The data set has three categorical variables (Sex, with levels Male and Female, Age, with levels Adult and Young, and Survival, with levels Alive and Dead) and nine numerical variables that hold various aspects of the birds’ anatomy, such as wingspread, weight, etc.
Problem 1: Make a logistic regression model that can predict survival status from all other predictor variables. (Include the categorical predictors Sex and Age.) Then do backwards selection, removing the predictors with the highest P value one by one, until you are only left with predictors that have P<0.1. How many and which predictors remain in the final model?
glm.out.complete <- glm(Survival ~ Sex + Age + Length + Wingspread + Weight + Skull_Length + Humerus_Length + Femur_Length + Tarsus_Length + Sternum_Length + Skull_Width, data = bumpus, family = "binomial")
summary(glm.out.complete)##
## Call:
## glm(formula = Survival ~ Sex + Age + Length + Wingspread + Weight +
## Skull_Length + Humerus_Length + Femur_Length + Tarsus_Length +
## Sternum_Length + Skull_Width, family = "binomial", data = bumpus)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.2342 -0.7890 -0.1887 0.7655 2.1927
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -10.79812 15.13435 -0.713 0.47555
## SexMale -1.64710 0.66562 -2.475 0.01334 *
## AgeYoung 0.32973 0.47216 0.698 0.48496
## Length 0.42375 0.10958 3.867 0.00011 ***
## Wingspread -0.01025 0.08496 -0.121 0.90394
## Weight 0.88472 0.24353 3.633 0.00028 ***
## Skull_Length -0.46347 0.46141 -1.004 0.31516
## Humerus_Length -1.66395 0.89997 -1.849 0.06447 .
## Femur_Length 0.09391 0.86933 0.108 0.91397
## Tarsus_Length -0.25479 0.39646 -0.643 0.52045
## Sternum_Length -0.67528 0.32942 -2.050 0.04037 *
## Skull_Width -0.68535 0.76052 -0.901 0.36750
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 188.07 on 135 degrees of freedom
## Residual deviance: 129.56 on 124 degrees of freedom
## AIC: 153.56
##
## Number of Fisher Scoring iterations: 5# remove Femur_Length
glm.out <- glm(Survival ~ Sex + Age + Length + Wingspread + Weight + Skull_Length + Humerus_Length + Tarsus_Length + Sternum_Length + Skull_Width, data = bumpus, family = "binomial")
summary(glm.out)##
## Call:
## glm(formula = Survival ~ Sex + Age + Length + Wingspread + Weight +
## Skull_Length + Humerus_Length + Tarsus_Length + Sternum_Length +
## Skull_Width, family = "binomial", data = bumpus)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.2444 -0.7987 -0.1872 0.7588 2.1838
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -10.90876 15.08451 -0.723 0.469571
## SexMale -1.65241 0.66405 -2.488 0.012833 *
## AgeYoung 0.32606 0.47066 0.693 0.488453
## Length 0.42440 0.10950 3.876 0.000106 ***
## Wingspread -0.01035 0.08493 -0.122 0.903042
## Weight 0.88092 0.24042 3.664 0.000248 ***
## Skull_Length -0.45503 0.45461 -1.001 0.316864
## Humerus_Length -1.61244 0.76223 -2.115 0.034393 *
## Tarsus_Length -0.23454 0.34926 -0.672 0.501877
## Sternum_Length -0.67692 0.32915 -2.057 0.039727 *
## Skull_Width -0.68140 0.75958 -0.897 0.369679
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 188.07 on 135 degrees of freedom
## Residual deviance: 129.57 on 125 degrees of freedom
## AIC: 151.57
##
## Number of Fisher Scoring iterations: 5# remove Wingspread
glm.out <- glm(Survival ~ Sex + Age + Length + Weight + Skull_Length + Humerus_Length + Tarsus_Length + Sternum_Length + Skull_Width, data = bumpus, family = "binomial")
summary(glm.out)##
## Call:
## glm(formula = Survival ~ Sex + Age + Length + Weight + Skull_Length +
## Humerus_Length + Tarsus_Length + Sternum_Length + Skull_Width,
## family = "binomial", data = bumpus)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.2472 -0.7907 -0.1847 0.7602 2.1922
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -11.7689 13.3246 -0.883 0.377102
## SexMale -1.6949 0.5657 -2.996 0.002736 **
## AgeYoung 0.3355 0.4641 0.723 0.469700
## Length 0.4197 0.1023 4.102 4.09e-05 ***
## Weight 0.8799 0.2402 3.663 0.000249 ***
## Skull_Length -0.4491 0.4517 -0.994 0.320046
## Humerus_Length -1.6458 0.7111 -2.315 0.020637 *
## Tarsus_Length -0.2429 0.3424 -0.709 0.478129
## Sternum_Length -0.6833 0.3251 -2.102 0.035534 *
## Skull_Width -0.6853 0.7583 -0.904 0.366162
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 188.07 on 135 degrees of freedom
## Residual deviance: 129.59 on 126 degrees of freedom
## AIC: 149.59
##
## Number of Fisher Scoring iterations: 5# remove Tarsus_Length
glm.out <- glm(Survival ~ Sex + Age + Length + Weight + Skull_Length + Humerus_Length + Sternum_Length + Skull_Width, data = bumpus, family = "binomial")
summary(glm.out)##
## Call:
## glm(formula = Survival ~ Sex + Age + Length + Weight + Skull_Length +
## Humerus_Length + Sternum_Length + Skull_Width, family = "binomial",
## data = bumpus)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.2465 -0.8113 -0.1847 0.7575 2.1017
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -10.7332 13.1537 -0.816 0.414509
## SexMale -1.5723 0.5368 -2.929 0.003403 **
## AgeYoung 0.3565 0.4624 0.771 0.440738
## Length 0.4173 0.1012 4.122 3.76e-05 ***
## Weight 0.8720 0.2393 3.645 0.000268 ***
## Skull_Length -0.5098 0.4416 -1.154 0.248299
## Humerus_Length -1.9455 0.5887 -3.305 0.000950 ***
## Sternum_Length -0.6725 0.3235 -2.079 0.037597 *
## Skull_Width -0.7033 0.7561 -0.930 0.352291
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 188.07 on 135 degrees of freedom
## Residual deviance: 130.08 on 127 degrees of freedom
## AIC: 148.08
##
## Number of Fisher Scoring iterations: 5# remove Age
glm.out <- glm(Survival ~ Sex + Length + Weight + Skull_Length + Humerus_Length + Sternum_Length + Skull_Width, data = bumpus, family = "binomial")
summary(glm.out)##
## Call:
## glm(formula = Survival ~ Sex + Length + Weight + Skull_Length +
## Humerus_Length + Sternum_Length + Skull_Width, family = "binomial",
## data = bumpus)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.2938 -0.7823 -0.1952 0.7758 2.0455
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -11.2855 12.9825 -0.869 0.384691
## SexMale -1.5916 0.5324 -2.990 0.002794 **
## Length 0.4220 0.1009 4.182 2.89e-05 ***
## Weight 0.8556 0.2350 3.641 0.000272 ***
## Skull_Length -0.5374 0.4378 -1.228 0.219594
## Humerus_Length -1.9022 0.5836 -3.259 0.001117 **
## Sternum_Length -0.6851 0.3207 -2.136 0.032643 *
## Skull_Width -0.6582 0.7511 -0.876 0.380807
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 188.07 on 135 degrees of freedom
## Residual deviance: 130.68 on 128 degrees of freedom
## AIC: 146.68
##
## Number of Fisher Scoring iterations: 5# remove Skull_Width
glm.out <- glm(Survival ~ Sex + Length + Weight + Skull_Length + Humerus_Length + Sternum_Length, data = bumpus, family = "binomial")
summary(glm.out)##
## Call:
## glm(formula = Survival ~ Sex + Length + Weight + Skull_Length +
## Humerus_Length + Sternum_Length, family = "binomial", data = bumpus)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.4048 -0.7911 -0.1888 0.7747 1.9636
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -15.6788 11.9179 -1.316 0.188319
## SexMale -1.5427 0.5285 -2.919 0.003511 **
## Length 0.4193 0.1011 4.149 3.34e-05 ***
## Weight 0.8319 0.2321 3.584 0.000339 ***
## Skull_Length -0.6294 0.4228 -1.488 0.136635
## Humerus_Length -1.9684 0.5770 -3.412 0.000646 ***
## Sternum_Length -0.7108 0.3205 -2.218 0.026585 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 188.07 on 135 degrees of freedom
## Residual deviance: 131.46 on 129 degrees of freedom
## AIC: 145.46
##
## Number of Fisher Scoring iterations: 5# remove Skull_Length
glm.out.final <- glm(Survival ~ Sex + Length + Weight + Humerus_Length + Sternum_Length, data = bumpus, family = "binomial")
summary(glm.out.final)##
## Call:
## glm(formula = Survival ~ Sex + Length + Weight + Humerus_Length +
## Sternum_Length, family = "binomial", data = bumpus)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.4921 -0.7678 -0.2155 0.7890 2.0192
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -23.15186 10.83789 -2.136 0.032663 *
## SexMale -1.39306 0.51054 -2.729 0.006360 **
## Length 0.38266 0.09487 4.034 5.49e-05 ***
## Weight 0.76098 0.22248 3.420 0.000625 ***
## Humerus_Length -2.17650 0.55596 -3.915 9.05e-05 ***
## Sternum_Length -0.75484 0.31296 -2.412 0.015870 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 188.07 on 135 degrees of freedom
## Residual deviance: 133.72 on 130 degrees of freedom
## AIC: 145.72
##
## Number of Fisher Scoring iterations: 5The final model uses five predictors, Sex, Length, Weight, Humerus_Length, and Sternum_Length.
Problem 2: Make a plot of the fitted probability as a function of the linear predictor, colored by survival. Make a density plot that shows how the two outcomes are separated by the linear predictor. Interperet your plots in 1-2 sentences. If you had to choose a cut-off value for alive or dead, where would it be?
reg_data <- data.frame(
predictor = glm.out.final$linear.predictors,
prob = glm.out.final$fitted.values,
Survival = bumpus$Survival
)
ggplot(reg_data, aes(x = predictor, y = prob, color = Survival)) + geom_point() + scale_color_colorblind()
ggplot(reg_data, aes(x = predictor, fill = Survival)) + geom_density(alpha = 0.5) + scale_fill_colorblind()
Our predictors do not cleanly separate the two survival outcomes alive and dead. There is no single line that can be drawn to separate survival outcomes.
Problem 3: Add rugs to both the top and bottom of the plot above. BONUS: Add a curve for the logistic function.
# extract data for alive and dead
alive_data <- filter(reg_data, Survival == "Alive")
dead_data <- filter(reg_data, Survival == "Dead")
# make data frame with logistic function spanning the minimum predictor value to the maximum
predictor <- seq(min(reg_data$predictor), max(reg_data$predictor), 0.1)
prob <- exp(predictor) / (1 + exp(predictor))
log_fun_data <- data.frame(predictor, prob)
# plot
ggplot(reg_data, aes(x = predictor, y = prob, color = Survival)) +
geom_line(data = log_fun_data, color = "black") +
geom_point() +
geom_rug(data = alive_data, sides = "b") +
geom_rug(data = dead_data, sides = "t") +
scale_color_colorblind()