12/1/2021

One test to rule them all

  • FLOWERS = SEX (ANOVA)
  • GPA = IQ (regression)
  • But both are RESPONSE (dependent) VARIABLE = EXPLANATORY (independent) VARIABLE
  • Only difference is SEX is categorical and IQ is continous

Last week’s ANOVA

anova(lm(FLOWERS~SEX, data = dioecious))
## Analysis of Variance Table
## 
## Response: FLOWERS
##           Df  Sum Sq Mean Sq F value Pr(>F)
## SEX        1  171841  171841  1.1754 0.2837
## Residuals 48 7017255  146193

Last week’s regression

summary(lm(gpa~iq, data = gpa_iq))
## 
## Call:
## lm(formula = gpa ~ iq, data = gpa_iq)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.3182 -0.5377  0.2178  1.0268  3.5785 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -3.55706    1.55176  -2.292   0.0247 *  
## iq           0.10102    0.01414   7.142 4.74e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.635 on 76 degrees of freedom
## Multiple R-squared:  0.4016, Adjusted R-squared:  0.3937 
## F-statistic: 51.01 on 1 and 76 DF,  p-value: 4.737e-10

Last week’s ANOVA (with coefficents)

options(contrasts=c('contr.sum','contr.poly')) #Whats' this about
summary(lm(FLOWERS~SEX, data = dioecious))
## 
## Call:
## lm(formula = FLOWERS ~ SEX, data = dioecious)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -430.4 -267.4 -118.5  194.3 1050.6 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   371.53      55.19   6.732 1.89e-08 ***
## SEX1          -59.83      55.19  -1.084    0.284    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 382.4 on 48 degrees of freedom
## Multiple R-squared:  0.0239, Adjusted R-squared:  0.003568 
## F-statistic: 1.175 on 1 and 48 DF,  p-value: 0.2837

The ANOVA table of last week’s regression

anova(lm(gpa~iq, data = gpa_iq))
## Analysis of Variance Table
## 
## Response: gpa
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## iq         1 136.32 136.319  51.008 4.737e-10 ***
## Residuals 76 203.11   2.672                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Aliassing (Contrasts)

  • When I reported the coefficients of an ANOVA analysis we used the grand mean as the intercept (sum contrasts e.g. contr.sum).
  • This is not the normal way, stats packages do, rather they use the mean of one of the levels (e.g. male), know as treatment contrasts (contr. treatment). This the default.
  • For teaching, we will often use the grand mean, but when you are doing your analysis, I would let R do its thing (unless the lecturer explicitly tells you not to.)

Last week’s ANOVA (with coefficents)

options(contrasts=c('contr.sum','contr.poly')) #Whats' this about
summary(lm(FLOWERS~SEX, data = dioecious))
## 
## Call:
## lm(formula = FLOWERS ~ SEX, data = dioecious)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -430.4 -267.4 -118.5  194.3 1050.6 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   371.53      55.19   6.732 1.89e-08 ***
## SEX1          -59.83      55.19  -1.084    0.284    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 382.4 on 48 degrees of freedom
## Multiple R-squared:  0.0239, Adjusted R-squared:  0.003568 
## F-statistic: 1.175 on 1 and 48 DF,  p-value: 0.2837

Last week’s ANOVA (with coefficents) ala R

options(contrasts=c('contr.treatment','contr.poly')) #Whats' this about
summary(lm(FLOWERS~SEX, data = dioecious))
## 
## Call:
## lm(formula = FLOWERS ~ SEX, data = dioecious)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -430.4 -267.4 -118.5  194.3 1050.6 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    311.7       85.5   3.646 0.000655 ***
## SEX2           119.7      110.4   1.084 0.283701    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 382.4 on 48 degrees of freedom
## Multiple R-squared:  0.0239, Adjusted R-squared:  0.003568 
## F-statistic: 1.175 on 1 and 48 DF,  p-value: 0.2837

Next lecture

  • More than one more explanatory variable (things are getting exciting now!)