Homework 6 — Linear and Logistic Regression
Cheslav Lukashanets
This is the last homework. Part 1 uses linear regression on country-level data. Part 2 uses logistic regression on a medical dataset.
Part 1 — Linear Regression: AllCountries dataset
Download AllCountries.csv from the Datasets folder on
Blackboard. The dataset has 217 countries with variables including GDP,
LifeExpectancy, Health, Internet, CO2, Energy, Electricity, and
more.
setwd("C:/Users/chesl/Desktop/DATA101/datasets")
countries <- read.csv("AllCountries.csv")
head(countries)## Country Code LandArea Population Density GDP Rural CO2 PumpPrice
## 1 Afghanistan AFG 652.86 37.172 56.9 521 74.5 0.29 0.70
## 2 Albania ALB 27.40 2.866 104.6 5254 39.7 1.98 1.36
## 3 Algeria DZA 2381.74 42.228 17.7 4279 27.4 3.74 0.28
## 4 American Samoa ASM 0.20 0.055 277.3 NA 12.8 NA NA
## 5 Andorra AND 0.47 0.077 163.8 42030 11.9 5.83 NA
## 6 Angola AGO 1246.70 30.810 24.7 3432 34.5 1.29 0.97
## Military Health ArmedForces Internet Cell HIV Hunger Diabetes BirthRate
## 1 3.72 2.01 323 11.4 67.4 NA 30.3 9.6 32.5
## 2 4.08 9.51 9 71.8 123.7 0.1 5.5 10.1 11.7
## 3 13.81 10.73 317 47.7 111.0 0.1 4.7 6.7 22.3
## 4 NA NA NA NA NA NA NA NA NA
## 5 NA 14.02 NA 98.9 104.4 NA NA 8.0 NA
## 6 9.40 5.43 117 14.3 44.7 1.9 23.9 3.9 41.3
## DeathRate ElderlyPop LifeExpectancy FemaleLabor Unemployment Energy
## 1 6.6 2.6 64.0 50.3 1.5 NA
## 2 7.5 13.6 78.5 55.9 13.9 808
## 3 4.8 6.4 76.3 16.4 12.1 1328
## 4 NA NA NA NA NA NA
## 5 NA NA NA NA NA NA
## 6 8.4 2.5 61.8 76.4 7.3 545
## Electricity Developed
## 1 NA NA
## 2 2309 1
## 3 1363 1
## 4 NA NA
## 5 NA NA
## 6 312 1Q1 — Simple Linear Regression
Fit a simple linear regression model predicting
LifeExpectancy from GDP.
# Your code:
lm1 <- lm(LifeExpectancy ~ GDP, data = countries)
summary(lm1)##
## Call:
## lm(formula = LifeExpectancy ~ GDP, data = countries)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.352 -3.882 1.550 4.458 9.330
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.842e+01 5.415e-01 126.36 <2e-16 ***
## GDP 2.476e-04 2.141e-05 11.56 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.901 on 177 degrees of freedom
## (38 observations deleted due to missingness)
## Multiple R-squared: 0.4304, Adjusted R-squared: 0.4272
## F-statistic: 133.7 on 1 and 177 DF, p-value: < 2.2e-16Report the intercept and slope. What does the slope mean in plain English (e.g., “for every X increase in GDP, life expectancy increases by Y”)?
intercept: 68.42, slope: 2.476e-4
For every 1 incresase in GDP, life expectancy increases by 2.476e-4.
What does the R² value tell you about how well GDP explains life expectancy?
Adjusted R-squared: 0.4272. 42.72% of the variance in countries’ life expectancy can be explained by its relationship with GDP.
Q2 — Multiple Linear Regression
Fit a multiple regression predicting LifeExpectancy from
GDP, Health, and Internet.
# Your code:
lm2 <- lm(LifeExpectancy ~ GDP + Health + Internet, data = countries)
summary(lm2)##
## Call:
## lm(formula = LifeExpectancy ~ GDP + Health + Internet, data = countries)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.5662 -1.8227 0.4108 2.5422 9.4161
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.908e+01 8.149e-01 72.499 < 2e-16 ***
## GDP 2.367e-05 2.287e-05 1.035 0.302025
## Health 2.479e-01 6.619e-02 3.745 0.000247 ***
## Internet 1.903e-01 1.656e-02 11.490 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.104 on 169 degrees of freedom
## (44 observations deleted due to missingness)
## Multiple R-squared: 0.7213, Adjusted R-squared: 0.7164
## F-statistic: 145.8 on 3 and 169 DF, p-value: < 2.2e-16Interpret the coefficient on Health (controlling
for GDP and Internet).
Health coeff: 2.479e-1. For every increase by 1 in Health alone, LifeExpectancy should increase by 2.479e-1.
How does the adjusted R² compare to the simple model in Q1? What does that suggest about adding predictors?
Adjusted R-squared: 0.7164, substantially larger than in the first model. This suggests that adding more predictors makes the model more accurate.
Q3 — Checking Assumptions
For the simple model in Q1 (LifeExpectancy ~ GDP):
- Briefly describe what you would CHECK to evaluate homoscedasticity and normality of residuals. What would an ideal outcome look like?
Check homoscedasticity (equal variance) with a residuals-fitted or scale-location plot. Ideally, the points should be spread evenly, and red line should be roughly horizontal.
Normality of residuals can be checked with a Q-Q plot, residuals should mostly follow the diagonal line.
- Then code your check (residual plot + Q-Q plot of residuals) and reflect on what you see.
# Your code:
plot(lm1, which = 1)
plot(lm1, which = 2)
Your reflection: Homoscedasticity is violated. The points are skewed right, and the red line has a noticeable curve, indicating unequal variance of residuals.
Q-Q plot also has a noticeable curve.
Q4 — Diagnosing Fit (RMSE)
For the multiple regression in Q2, calculate the RMSE (root mean squared error).
sqrt(mean(residuals(lm2)^2))## [1] 4.056417What does the RMSE represent in the context of predicting life expectancy? How would large residuals for certain countries affect your confidence in the model?
RMSE represents the average error of the model. Large residuals increase the mean, increasing the average error. Large residuals are also less easily explained by variance (?), decreasing confidence that the model is good at predicting the value.
Q5 — Multicollinearity (no code)
Suppose you fit a regression predicting CO2 using both
Energy and Electricity. These two predictors
are highly correlated.
Explain in 2-3 sentences how this multicollinearity could affect (a) the interpretation of the coefficients and (b) the reliability of the model.
Multicollinearity is confusing for the model, as it indicates that >=two vars have overlapping effects. This can weaken the model’s success rates, and makes it that much more difficult to isolate the meaningful vars from unimportant noise.
Part 2 — Logistic Regression: Pima Indians Diabetes
This part uses the Pima Indians Diabetes dataset (768 patients, binary outcome: 0 = no diabetes, 1 = diabetes).
Don’t change this chunk — it loads and cleans the data:
url <- "https://raw.githubusercontent.com/jbrownlee/Datasets/master/pima-indians-diabetes.data.csv"
data <- read.csv(url, header = FALSE)
colnames(data) <- c("Pregnancies", "Glucose", "BloodPressure", "SkinThickness",
"Insulin", "BMI", "DiabetesPedigreeFunction", "Age", "Outcome")
data$Outcome <- as.factor(data$Outcome)
# Replace impossible 0 values with NA
data$Glucose[data$Glucose == 0] <- NA
data$BloodPressure[data$BloodPressure == 0] <- NA
data$BMI[data$BMI == 0] <- NA
colSums(is.na(data))## Pregnancies Glucose BloodPressure
## 0 5 35
## SkinThickness Insulin BMI
## 0 0 11
## DiabetesPedigreeFunction Age Outcome
## 0 0 0Q6 — Fit the Logistic Model
Fit a logistic regression predicting Outcome from
Glucose, BMI, and Age.
glm1 <- glm(Outcome ~ Glucose + BMI + Age, data = data, family = "binomial")Get the summary of the model. For each predictor, does an increase RAISE or LOWER the odds of diabetes? Which predictors are significant (p < 0.05)?
summary(glm1)##
## Call:
## glm(formula = Outcome ~ Glucose + BMI + Age, family = "binomial",
## data = data)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -9.032377 0.711037 -12.703 < 2e-16 ***
## Glucose 0.035548 0.003481 10.212 < 2e-16 ***
## BMI 0.089753 0.014377 6.243 4.3e-10 ***
## Age 0.028699 0.007809 3.675 0.000238 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 974.75 on 751 degrees of freedom
## Residual deviance: 724.96 on 748 degrees of freedom
## (16 observations deleted due to missingness)
## AIC: 732.96
##
## Number of Fisher Scoring iterations: 4Every predictor has a positive (estimate > 0) and significant (p-value < 0.05) correlation with odds for diabetes.
Q7 — Confusion Matrix
Use threshold 0.5 to convert predicted probabilities into 0/1 predictions, then build a confusion matrix.
data$pred_prob <- predict(glm1, data, type = "response")
data$pred_class <- ifelse(data$pred_prob > 0.5, 1, 0)
table(Actual = data$Outcome, Predicted = data$pred_class)## Predicted
## Actual 0 1
## 0 429 59
## 1 114 150Report the confusion matrix counts: TP, TN, FP, FN. True positive: 150, True Negative: 429, False Positive: 59, False Negative: 114. —
Q8 — Accuracy, Precision, Recall
From your confusion matrix, calculate accuracy, precision, and recall.
accuracy = (150+429)/(150+429+59+114)
precision = 150/(150+59)
recall = 150/(150+114)Report all three values. In a medical screening context, which is more important — precision or recall? Why?
accuracy = 0.770 precision = 0.718 recall = 0.568
In a medical screening context, recall is more important, because it is better to have a false positive (healthy patient flagged as sick) than to miss a sick patient.
Q9 — ROC and AUC
Plot the ROC curve and compute the AUC.
# install.packages("pROC") if needed
library(pROC)## Warning: package 'pROC' was built under R version 4.5.3## Type 'citation("pROC")' for a citation.##
## Attaching package: 'pROC'## The following objects are masked from 'package:stats':
##
## cov, smooth, var# Your code:
roc_obj <- roc(data$Outcome, data$pred_class)## Setting levels: control = 0, case = 1## Setting direction: controls < casesplot(roc_obj)
auc(roc_obj)## Area under the curve: 0.7236Report the AUC. Is your model closer to random guessing (AUC = 0.5) or perfect (AUC = 1)? Describe its overall performance in one sentence.
AUC: 0.7236
The model is closer to random guessing than perfect guessing, but only by a little.