Combining (lots of) numerical and categorical predictors

ANCOVAs and beyond

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Overview

This practice reviews the lecture on combining numerical and continous predictors.

Example

Following the iris example from class

set.seed(3)
iris_example_species <-data.frame(
  Species = c(rep("baruch",25), rep("hunter", 25), rep("york", 25)),
  Petal_Length = runif(75,2,4 ))
set.seed(31)
iris_example_species$Sepal_interaction <- 
  iris_example_species$Petal_Length * c(rep(-2, 25),rep(0,25), rep(5,25)) + 
  c(rep(2,25), rep(3,25), rep(4,25)) + rnorm(75)

Plot the data

library(ggplot2)
ggplot(iris_example_species, aes(x= Petal_Length, y = Sepal_interaction, color = Species)) +
  geom_point()+
  ylab("Sepal Length") +
  xlab("Petal Length") +
  ggtitle("Impact of petal length and species on sepal length") +
    geom_smooth(method = "lm", se = F)

`geom_smooth()` using formula = 'y ~ x'

Analysis would indicate (assumption plots not shown here to allow focus on interpreting interactions)

library(car)

Warning: package 'car' was built under R version 4.4.1

Loading required package: carData

Anova(lm( Sepal_interaction~ Petal_Length * Species, iris_example_species), 
      type = "III")

Anova Table (Type III tests)

Response: Sepal_interaction
                      Sum Sq Df  F value    Pr(>F)    
(Intercept)            7.076  1   8.0267  0.006038 ** 
Petal_Length          38.452  1  43.6177  6.88e-09 ***
Species                3.353  2   1.9015  0.157092    
Petal_Length:Species 227.334  2 128.9368 < 2.2e-16 ***
Residuals             60.828 69                       
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

interactions do exist. This means we can’t interpret the “general” relationship, so we need to look for each species using regression.

summary(lm(Sepal_interaction ~ Petal_Length, 
         iris_example_species[iris_example_species$Species == "baruch",]))


Call:
lm(formula = Sepal_interaction ~ Petal_Length, data = iris_example_species[iris_example_species$Species == 
    "baruch", ])

Residuals:
    Min      1Q  Median      3Q     Max 
-1.7194 -0.5504 -0.1860  0.4736  1.7067 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)    2.9734     0.9767   3.044  0.00576 ** 
Petal_Length  -2.3663     0.3335  -7.097 3.14e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.8738 on 23 degrees of freedom
Multiple R-squared:  0.6865,    Adjusted R-squared:  0.6728 
F-statistic: 50.36 on 1 and 23 DF,  p-value: 3.144e-07

Anova(lm(Sepal_interaction ~ Petal_Length, 
         iris_example_species[iris_example_species$Species == "baruch",]), 
      type="III")

Anova Table (Type III tests)

Response: Sepal_interaction
             Sum Sq Df F value    Pr(>F)    
(Intercept)   7.076  1  9.2676  0.005758 ** 
Petal_Length 38.452  1 50.3604 3.144e-07 ***
Residuals    17.561 23                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

summary(lm(Sepal_interaction ~ Petal_Length, 
           iris_example_species[iris_example_species$Species == "hunter",]))


Call:
lm(formula = Sepal_interaction ~ Petal_Length, data = iris_example_species[iris_example_species$Species == 
    "hunter", ])

Residuals:
     Min       1Q   Median       3Q      Max 
-1.89221 -0.58055  0.00876  0.47006  2.49756 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)
(Intercept)    1.2564     1.1463   1.096    0.284
Petal_Length   0.4962     0.3895   1.274    0.215

Residual standard error: 0.9902 on 23 degrees of freedom
Multiple R-squared:  0.06589,   Adjusted R-squared:  0.02528 
F-statistic: 1.622 on 1 and 23 DF,  p-value: 0.2155

Anova(lm(Sepal_interaction ~ Petal_Length, 
         iris_example_species[iris_example_species$Species == "hunter",]), 
      type="III")

Anova Table (Type III tests)

Response: Sepal_interaction
              Sum Sq Df F value Pr(>F)
(Intercept)   1.1779  1  1.2014 0.2844
Petal_Length  1.5907  1  1.6224 0.2155
Residuals    22.5503 23

summary(lm(Sepal_interaction ~ Petal_Length, 
           iris_example_species[iris_example_species$Species == "york",]))


Call:
lm(formula = Sepal_interaction ~ Petal_Length, data = iris_example_species[iris_example_species$Species == 
    "york", ])

Residuals:
     Min       1Q   Median       3Q      Max 
-1.45150 -0.69660  0.02717  0.83006  1.64698 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)    4.0617     0.9550   4.253    3e-04 ***
Petal_Length   4.9642     0.3024  16.417  3.4e-14 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.9491 on 23 degrees of freedom
Multiple R-squared:  0.9214,    Adjusted R-squared:  0.918 
F-statistic: 269.5 on 1 and 23 DF,  p-value: 3.401e-14

Anova(lm(Sepal_interaction ~ Petal_Length, 
         iris_example_species[iris_example_species$Species == "york",]), 
      type="III")

Anova Table (Type III tests)

Response: Sepal_interaction
              Sum Sq Df F value    Pr(>F)    
(Intercept)   16.292  1  18.087 0.0002998 ***
Petal_Length 242.770  1 269.527 3.401e-14 ***
Residuals     20.717 23                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Here we see that there is a significant negative relationship (F_1,23 = 50.36, p<0.001) between sepal and petal length for I. baruch, a significant positive relationship (F_1,23 = 269.53, p<0.001) between sepal and petal length for I. york,and no relationship (F_1,23 = 1.63, p<-0.21) between sepal and petal length for I. hunter.

Practice

1

Data on FEV (forced expiratory volume), a measure of lung function, can be found at

http://www.statsci.org/data/general/fev.txt

More information on the dataset is available at

http://www.statsci.org/data/general/fev.html.

Does the impact of age on FEV differ among genders? Consider how your answer to this differs from the previous assignment!

2

Data on home gas consumption at various temperatures before and after new insulation was installed has been collected @

http://www.statsci.org/data/general/insulgas.txt

More information on the data is available @

http://www.statsci.org/data/general/insulgas.html

Is there any relationship between these factors? How would you test this, and what type of plot would you produce to accompany your analysis?

3

Data on the height, diameter, and volume of cherry trees was collected for use in developing an optimal model to predict timber volume. Data is available @

http://www.statsci.org/data/general/cherry.txt

Use the data to justify an optimal model.

4

Over the course of five years, a professor asked students in his stats class to carry out a simple experiment. Students were asked to measure their pulse rate, run for one minute, then measure their pulse rate again. The students also filled out a questionnaire. Data include:

Variable	Description
Height	Height (cm)
Weight	Weight (kg)
Age	Age (years)
Gender	Sex (1 = male, 2 = female)
Smokes	Regular smoker? (1 = yes, 2 = no)
Alcohol	Regular drinker? (1 = yes, 2 = no)
Exercise	Frequency of exercise (1 = high, 2 = moderate, 3 = low)
Change	Percent change in pulse (pulse after experiment/pulse before experiment)
Year	Year of class (93 - 98)

Using the available data (available at

https://docs.google.com/spreadsheets/d/e/2PACX-1vToN77M80enimQglwpFroooLzDtcQMh4qKbOuhbu-eVmU9buczh7nVV1BdI4T_ma-PfWUnQYmq-60RZ/pub?gid=942311716&single=true&output=csv )

determine the optimal subset of explanatory variables that should be used to predict change pulse rate (Change) (focusing on main effects only, no interactions) and explain your choice of methods. Interpret your results. Make sure you can explain any changes you needed to make to the dataset or steps you used in your analysis.

5

Find one example of model selection from a paper in your field or area of interest. It may be more complicated (see next question!) than what we have done, but try to identify the approach (F/AIC, top-down/bottom-up/not nested) they used. Review how they explained their approach (methods) and reported outcomes (results). Be prepared to discuss in class next week.

6 (78002 only)

Find one example of a linear model selection (e.g., generalized linear models, mixed-effects models, beta regression) from a paper in your field. Be prepared to name the technique in class next week.