Lecture 9 - Simple Linear Regression
# Topics for today! 1. Topic: Checking correlation 2. Topic: Simple Linear Regression Basics 3. Topic: Checking the assumptions of linear regression 4. Topic: Plotting simple linear regression 5. Topic: Linear Regression Example in R ## Data ```{r} setwd("~/Desktop/R Materials/mih140/Lecture 9 - Linear Regression I") house = read.table("house.txt", sep = "\t", header = T) # quick pruning of outliers house = subset(house, price >= (mean(price) - 3*sd(price)) & price <= (mean(price) + 3*sd(price))) ``` ## 1. Topic: Checking correlation ```{r} # Test if correlation between price and footage is nonzero. cor.test(house$price, house$sqft_living) cor(house[,c("price", "sqft_living")]) # Compute the correlation itself ``` ### Note correlation being zero does not imply no relationship, just no linear relationship ```{r} x = cos(seq(0, 2*pi, .1)) y = sin(seq(0, 2*pi, .1)) plot(x,y) cor(x,y) # Basically 0, although clearly related! ``` # 2. Topic: Simple Linear Regression Basics ```{r} # Lets use lin reg to see the relationship between price and living space reg_model = lm(data = house, price ~ sqft_living) # Simple linear regression reg_model_2 = lm(data = house, price ~ 0 + sqft_living) # Simple linear regression through the origin # Examining the regression reg_model$coefficients # gets the coeff. of the model reg_model$fitted.values # what the model predicts for each point avg_error = mean(abs(reg_model$fitted.values - house$price)) # same as mean(abs(reg_model$residuals)) summary(reg_model) # Can read off if coeff. are stat. sign. diff. from 0 ``` ## 3. Topic: Checking the assumptions of linear regression ```{r} # The assumptions are: ## 1. Linearity - Look at residuals ## 2. Normality - qqplot of residuals ## 3. Independence (of error term and response) - Next time ## 4. Constant Variance - Look at residuals # Check assumptions by examining par(mfrow = c(2,2)) plot(reg_model) # If the assumptions are satisfied, we can do inference using our model. More next time! ``` ## 4. Topic: Plotting simple linear regression ```{r} # Plotting code from before price_lim = c(min(house$price), max(house$price)) sqrtft_lim = c(min(house$sqft_living), max(house$sqft_living)) plot(data = house[house$bedrooms <= 3,], price ~ sqft_living, col = "red", xlim = sqrtft_lim, ylim = price_lim) par(new = T) # tells R to hold prev plot plot(data = house[house$bedrooms > 3,], price ~ sqft_living, col = "blue", xlim = sqrtft_lim, ylim = price_lim) # Adding lines to plots abline(reg_model, col = "black", lwd = 2) # Adds a regression line abline(0,200, lwd = 2, col = "green") # Adds a second regression line abline(reg_model_2, lwd = 2, col = "black") # Adds a third regression line ``` ## Topic: Linear Regression Example in R ```{r} setwd("~/Desktop/R Materials/mih140/Lecture 10 - Linear Regression I") cba = read.table("cba_admissions_1999.txt", sep = "\t", header = T, quote = "", allowEscapes = T) cba = cba[!is.na(cba$SAT_math),] # remove na rows ``` ```{r} # In this example we examine the relationship betwee sat_math score and sat_verbal score in the cba dataset. # 1: Fit a simple regression between SAT_verbal and SAT_math res = lm(cba$SAT_verbal ~ cba$SAT_math) # 2: Write out the regression equation summary(res) # observe model Y = 0.46625 X + 290.08827 # 3: Plot the regression line against the scatter plot, be sure to appropriately label your plot! plot(cba$SAT_math, cba$SAT_verbal, main = "Linear Relationship Between SAT Math and SAT Verbal", xlab = "SAT Math Score", ylab = "SAT Verbal Score") abline(res, col = "red") # 4: Plot the residuals, qqplot, comment on trends you see in the data and check assumptions par(mfrow = c(2,2)) plot(res) # Residuals seem relatively consistent, the equal variance assumption and independence between our independent variable and the normal noise assumption appears reasonable. # QQ Plot is straight and diagonal supporting our normality assumption ```
