19 Day 19 (June 28)

19.1 Announcements

19.2 Regression and ANOVA

  • Review of the linear model
  • Regression, ANOVA, and t-test as special cases of the linear model
    • Write the regression and ANOVA model out on white board
  • ASA statement on use of p-values (link)

19.3 T-test

  • Common statistical test taught in introductory statistics classes

    • Linear model representation

    • Example: comparing two means

      library(faraway)
df.plant <- PlantGrowth[-c(1:10),]
df.plant$group <- factor(df.plant$group) 

boxplot(weight ~ group,df.plant,col="grey",ylim=c(0,7))

      aggregate(weight ~ group, FUN=mean,data=df.plant)
      ##   group weight
## 1  trt1  4.661
## 2  trt2  5.526

    • Use t.test() function in R.

      t.test(weight ~ group-1,data = df.plant,var.equal = TRUE)
      ## 
##  Two Sample t-test
## 
## data:  weight by group
## t = -3.0101, df = 18, p-value = 0.007518
## alternative hypothesis: true difference in means between group trt1 and group trt2 is not equal to 0
## 95 percent confidence interval:
##  -1.4687336 -0.2612664
## sample estimates:
## mean in group trt1 mean in group trt2 
##              4.661              5.526

    • Use lm() function in R.

      # With "intercept"
m1 <- lm(weight ~ group,data = df.plant)
summary(m1)
      ## 
## Call:
## lm(formula = weight ~ group, data = df.plant)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.0710 -0.3573 -0.0910  0.2402  1.3690 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   4.6610     0.2032   22.94 8.93e-15 ***
## grouptrt2     0.8650     0.2874    3.01  0.00752 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6426 on 18 degrees of freedom
## Multiple R-squared:  0.3348, Adjusted R-squared:  0.2979 
## F-statistic: 9.061 on 1 and 18 DF,  p-value: 0.007518
      # Without "intercept"
m2 <- lm(weight ~ group-1,data = df.plant)
summary(m2)
      ## 
## Call:
## lm(formula = weight ~ group - 1, data = df.plant)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.0710 -0.3573 -0.0910  0.2402  1.3690 
## 
## Coefficients:
##           Estimate Std. Error t value Pr(>|t|)    
## grouptrt1   4.6610     0.2032   22.94 8.93e-15 ***
## grouptrt2   5.5260     0.2032   27.20 4.52e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6426 on 18 degrees of freedom
## Multiple R-squared:  0.986,  Adjusted R-squared:  0.9844 
## F-statistic: 632.9 on 2 and 18 DF,  p-value: < 2.2e-16
  • Live example