# Software Help 8

Software Help 8

The next two pages cover the  Minitab and   commands for the procedures in this lesson.

Below is a zip file that contains all the data sets used in this lesson:

• birthsmokers.txt
• birthsmokers_02.txt
• depression.txt
• shipment

# Minitab Help 8: Categorical Predictors

Minitab Help 8: Categorical Predictors

## Birthweight and smoking (2-level categorical predictor, additive model)

• Create an indicator variable for Smoke by selecting Calc > Make Indicator Variables, move "Smoke" to the box labeled "Indicator variables for," and click "OK."
• Perform a linear regression analysis of Wgt on Gest + Smoke_yes. You can either put "Gest" and "Smoke_yes" in the box labeled "Continuous predictors" or, alternatively, put "Gest" in the box labeled "Continuous predictors" and "Smoke" in the box labeled "Categorical predictors." Either way, click "Storage" and select "Fits" before clicking "OK."
• Select Calc > Calculator, type "FITS0" in the box labeled "Store result in variable," type "if(Smoke="no",FITS)" in the box labeled "Expression," and click "OK." Repeat to create FITS1 as "if(Smoke="yes",FITS)."
• Create a basic scatterplot but select "With Groups" instead of "Simple." Plot "Wgt" vs "Gest" with "Smoke" as the "Categorical variable for grouping."
• To add parallel regression lines representing Smoke=0 and Smoke=1 to the scatterplot, select the scatterplot, select Editor > Add > Calculated Line, and select "FITS0" for the "Y column" and "Gest" for the "X column." Repeat to add the "FITS1" line.
• To display confidence intervals for the regression parameters, click "Results" in the Regression Dialog and select "Expanded tables" for "Display of results."
• Find a confidence interval and a prediction interval for the response to display confidence intervals for expected Wgt at Gest=38 (for Smoke=1 and Smoke=0).
• Select Data > Split Worksheet to create separate worksheets for Smoke=0 and Smoke=1.
• To repeat analysis using (1, -1) coding, click "Coding" in the Regression Dialog and select "(-1, 0, +1)" as the "Coding for categorical predictors."

## Depression treatment (3-level categorical predictor, interaction model)

• Create a basic scatterplot but select "With Groups" instead of "Simple." Plot "y" (treatment effectiveness) vs "age" with "TRT" as the "Categorical variable for grouping."
• Perform a linear regression analysis of y on age (continuous) + TRT (categorical) but before clicking "OK," click "Model," select age and TRT together in the "Predictors" box, change "Interactions through order" to "2" and click "Add." You should see "age*TRT" appear in the box labeled "Terms in the model." Before clicking "OK," click "Coding," select "(1, 0)" for the "Coding for categorical predictors," and change the "Reference level" to "C."
• To add non-parallel regression lines representing each of the three treatments to the scatterplot do the following. Create a basic scatterplot but select "With Regression and Groups" instead of "Simple." Plot "y" (treatment effectiveness) vs "age" with "TRT" as the "Categorical variable for grouping."
• Create residual plots and select "Residuals versus fits" (with regular residuals).
• Conduct regression error normality tests and select Anderson-Darling.
• Click "Options" in the regression dialog to select Sequential (Type I) sums of squares to display an Anova table allowing calculation of F-statistic to see if at least one of x2, x3, age.x2, and age.x3 are useful (i.e., the regression functions differ).
• Use the same Anova table to calculate an F-statistic to see if at least one of age.x2 and age.x3 are useful (i.e., the regression functions have different slopes).

## Real estate air conditioning (2-level categorical predictor, interaction model, transformations)

• Perform a linear regression analysis of SalePrice on SqFeet and Air (both continuous) but before clicking "OK," click "Model," select SqFeet and Air together in the "Predictors" box, change "Interactions through order" to "2" and click "Add." You should see "SqFeet*Air" appear in the box labeled "Terms in the model."
• To display a scatterplot of SalePrice vs SqFeet with points marked by Air and non-parallel regression lines representing Air=0 and Air=1 do the following. Create a basic scatterplot but select "With Regression and Groups" instead of "Simple." Plot "SalePrice" vs "SqFeet" with "Air" as the "Categorical variable for grouping."
• Create residual plots and select "Residuals versus fits" (with regular residuals).
• Select Calc > Calculator to create log(SalePrice) and log(SqFeet) variables and repeat preceding instructions to fit a multiple linear regression model of log(SalePrice) on log(SqFeet) + Air + log(SqFeet).Air.
• Repeat the preceding instructions to display a scatterplot of log(SalePrice) vs log(SqFeet) with points marked by Air and non-parallel regression lines representing Air=0 and Air=1.
• Create residual plots and select "Residuals versus fits" (with regular residuals).

## Hospital infection risk (4-level categorical predictor, additive model)

• Use Data > Subset Worksheet to select only hospitals with Stay < 14.
• Perform a linear regression analysis of InfctRsk on Stay and Xray (continuous) and Region (categorical) but before clicking "OK," click "Coding," select "(1, 0)" for the "Coding for categorical predictors," and change the "Reference level" to "1."
• The resulting Anova table displays an F-statistic to see if at least one of i2, i3, and i4 is useful (conclusion: the regression functions differ by region).
• To calculate an F-statistic to see if at least one of i2 and i3 are useful you'll need to first create indicator variables for Region by selecting Calc > Make Indicator Variables. To find the reduced model results, Perform a linear regression analysis of InfctRsk on Stay, Xray, and Region_4 (all continuous).

# R Help 8: Categorical Predictors

R Help 8: Categorical Predictors

## R

### Birthweight and smoking (2-level categorical predictor, additive model)

• Create a scatterplot matrix of the data
• Fit a multiple linear regression model of Wgt on Gest + Smoke.
• Display scatterplot of Wgt vs Gest with points marked by Smoke and add parallel regression lines representing Smoke=0 and Smoke=1.
• Display regression results and calculate confidence intervals for the regression parameters.
• Display confidence intervals for expected Wgt at Gest=38 (for Smoke=1 and Smoke=0).
• Repeat analysis separately for Smoke=0 and Smoke=1.
• Repeat analysis using (1, -1) coding.
birthsmokers <- read.table("~/path-to-folder/birthsmokers.txt", header=T)
attach(birthsmokers)

pairs(cbind(Wgt, Gest, Smoke))

model <- lm(Wgt ~ Gest + Smoke)

plot(x=Gest, y=Wgt, ylim=c(2300, 3700),
col=ifelse(Smoke=="yes", "red", "blue"),
panel.last = c(lines(sort(Gest[Smoke=="no"]),
fitted(model)[Smoke=="no"][order(Gest[Smoke=="no"])],
col="blue"),
lines(sort(Gest[Smoke=="yes"]),
fitted(model)[Smoke=="yes"][order(Gest[Smoke=="yes"])],
col="red")))

summary(model)
#              Estimate Std. Error t value Pr(>|t|)
# (Intercept) -2389.573    349.206  -6.843 1.63e-07 ***
# Gest          143.100      9.128  15.677 1.07e-15 ***
# Smokeyes     -244.544     41.982  -5.825 2.58e-06 ***
# ---
# Residual standard error: 115.5 on 29 degrees of freedom
# Multiple R-squared:  0.8964,  Adjusted R-squared:  0.8892
# F-statistic: 125.4 on 2 and 29 DF,  p-value: 5.289e-15

confint(model)
#                  2.5 %     97.5 %
# (Intercept) -3103.7795 -1675.3663
# Gest          124.4312   161.7694
# Smokeyes     -330.4064  -158.6817

predict(model, interval="confidence",
newdata=data.frame(Gest=c(38, 38), Smoke=c("yes", "no")))
#        fit      lwr      upr
# 1 2803.693 2740.599 2866.788
# 2 3048.237 2989.120 3107.355

model.0 <- lm(Wgt ~ Gest, subset=Smoke=="no")
summary(model.0)
#             Estimate Std. Error t value Pr(>|t|)
# (Intercept) -2546.14     457.29  -5.568 6.93e-05 ***
# Gest          147.21      11.97  12.294 6.85e-09 ***

predict(model.0, interval="confidence",
newdata=data.frame(Gest=38))
#        fit      lwr     upr
# 1 3047.724 2990.298 3105.15

model.1 <- lm(Wgt ~ Gest, subset=Smoke=="yes")
summary(model.1)
#             Estimate Std. Error t value Pr(>|t|)
# (Intercept) -2474.56     553.97  -4.467 0.000532 ***
# Gest          139.03      14.11   9.851 1.12e-07 ***

predict(model.1, interval="confidence",
newdata=data.frame(Gest=38))
#        fit      lwr      upr
# 1 2808.528 2731.726 2885.331

Smoke2 <- ifelse(Smoke=="yes", 1, -1)
model.3 <- lm(Wgt ~ Gest + Smoke2)
summary(model.3)
#              Estimate Std. Error t value Pr(>|t|)
# (Intercept) -2511.845    353.449  -7.107 8.07e-08 ***
# Gest          143.100      9.128  15.677 1.07e-15 ***
# Smoke2       -122.272     20.991  -5.825 2.58e-06 ***

# Alternatively
model.3 <- lm(Wgt ~ Gest + Smoke, contrasts=list(Smoke="contr.sum"))
summary(model.3)

detach(birthsmokers)

### Depression treatments (3-level categorical predictor, interaction model)

• Display scatterplot of y (treatment effectiveness) vs age with points marked by treatment.
• Create interaction variables and fit a multiple linear regression model of y on age + x2 + x3 + age.x2 + age.x3.
• Add non-parallel regression lines representing each of the three treatments to the scatterplot.
• Display a residuals vs fits plot and a normal probability plot of the residuals, and conduct an Anderson-Darling normality test using the nortest package.
• Conduct an F-test to see if at least one of x2, x3, age.x2, and age.x3 are useful (i.e., the regression functions differ).
• Conduct an F-test to see if at least one of age.x2 and age.x3 are useful (i.e., the regression functions have different slopes).
depression <- read.table("~/path-to-folder/depression.txt", header=T)
attach(depression)

plot(x=age, y=y, col=as.numeric(TRT))
legend("topleft", col=1:3, pch=1,
inset=0.02, x.intersp = 1.5, y.intersp = 1.8,
legend=c("Trt A", "Trt B", "Trt C"))

age.x2 <- age*x2
age.x3 <- age*x3

model.1 <- lm(y ~ age + x2 + x3 + age.x2 + age.x3)
summary(model.1)
#             Estimate Std. Error t value Pr(>|t|)
# (Intercept)  6.21138    3.34964   1.854 0.073545 .
# age          1.03339    0.07233  14.288 6.34e-15 ***
# x2          41.30421    5.08453   8.124 4.56e-09 ***
# x3          22.70682    5.09097   4.460 0.000106 ***
# age.x2      -0.70288    0.10896  -6.451 3.98e-07 ***
# age.x3      -0.50971    0.11039  -4.617 6.85e-05 ***

plot(x=age, y=y, ylim=c(20, 80), col=as.numeric(TRT),
panel.last = c(lines(sort(age[TRT=="A"]),
fitted(model.1)[TRT=="A"][order(age[TRT=="A"])],
col=1),
lines(sort(age[TRT=="B"]),
fitted(model.1)[TRT=="B"][order(age[TRT=="B"])],
col=2),
lines(sort(age[TRT=="C"]),
fitted(model.1)[TRT=="C"][order(age[TRT=="C"])],
col=3)))
legend("topleft", col=1:3, pch=1, lty=1,
inset=0.02, x.intersp = 1.5, y.intersp = 1.8,
legend=c("Trt A", "Trt B", "Trt C"))

plot(x=fitted(model.1), y=rstudent(model.1),
panel.last = abline(h=0, lty=2))

qqnorm(residuals(model.1), main="", datax=TRUE)
qqline(residuals(model.1), datax=TRUE)

library(nortest)
ad.test(residuals(model.1)) # A = 0.4057, p-value = 0.3345

anova(model.1)
#           Df Sum Sq Mean Sq  F value    Pr(>F)
# age        1 3424.4  3424.4 222.2946 2.059e-15 ***
# x2         1  803.8   803.8  52.1784 4.857e-08 ***
# x3         1    1.2     1.2   0.0772    0.7831
# age.x2     1  375.0   375.0  24.3430 2.808e-05 ***
# age.x3     1  328.4   328.4  21.3194 6.850e-05 ***
# Residuals 30  462.1    15.4

model.2 <- lm(y ~ age)
anova(model.2, model.1)
# Model 1: y ~ age
# Model 2: y ~ age + x2 + x3 + age.x2 + age.x3
#   Res.Df     RSS Df Sum of Sq     F    Pr(>F)
# 1     34 1970.57
# 2     30  462.15  4    1508.4 24.48 4.458e-09 ***

model.3 <- lm(y ~ age + x2 + x3)
anova(model.3, model.1)
# Model 1: y ~ age + x2 + x3
# Model 2: y ~ age + x2 + x3 + age.x2 + age.x3
#   Res.Df     RSS Df Sum of Sq      F   Pr(>F)
# 1     32 1165.57
# 2     30  462.15  2    703.43 22.831 9.41e-07 ***

detach(depression)

### Real estate air conditioning (2-level categorical predictor, interaction model, transformations)

• Create an interaction variable and fit a multiple linear regression model of SalePrice on SqFeet + Air + SqFeet.Air.
• Display scatterplot of SalePrice vs SqFeet with points marked by Air and add non-parallel regression lines representing Air=0 and Air=1.
• Display residual plot with fitted (predicted) values on the horizontal axis.
• Create log(SalePrice), log(SqFeet), and log(SqFeet).Air variables and fit a multiple linear regression model of log(SalePrice) on log(SqFeet) + Air + log(SqFeet).Air.
• Display scatterplot of log(SalePrice) vs log(SqFeet) with points marked by Air and add non-parallel regression lines representing Air=0 and Air=1.
• Display residual plot with fitted (predicted) values on the horizontal axis.
realestate <- read.table("~/path-to-folder/realestate.txt", header=T)
attach(realestate)

SqFeet.Air <- SqFeet*Air
model.1 <- lm(SalePrice ~ SqFeet + Air + SqFeet.Air)
summary(model.1)
#             Estimate Std. Error t value Pr(>|t|)
# (Intercept)   -3.218     30.085  -0.107 0.914871
# SqFeet       104.902     15.748   6.661 6.96e-11 ***
# Air          -78.868     32.663  -2.415 0.016100 *
# SqFeet.Air    55.888     16.580   3.371 0.000805 ***

plot(x=SqFeet, y=SalePrice,
col=ifelse(Air, "red", "blue"),
panel.last = c(lines(sort(SqFeet[Air==0]),
fitted(model.1)[Air==0][order(SqFeet[Air==0])],
col="blue"),
lines(sort(SqFeet[Air==1]),
fitted(model.1)[Air==1][order(SqFeet[Air==1])],
col="red")))
legend("topleft", col=c("blue", "red"), pch=1, lty=1, inset=0.02,
legend=c("No air conditioning", "Air conditioning"))

plot(x=fitted(model.1), y=residuals(model.1),
xlab="Fitted values", ylab="Residuals",
panel.last = abline(h=0, lty=2))

lnSalePrice <- log(SalePrice)
lnSqFeet <- log(SqFeet)
lnSqFeet.Air <- lnSqFeet*Air

model.2 <- lm(lnSalePrice ~ lnSqFeet + Air + lnSqFeet.Air)

plot(x=lnSqFeet, y=lnSalePrice,
col=ifelse(Air, "red", "blue"),
panel.last = c(lines(sort(lnSqFeet[Air==0]),
fitted(model.2)[Air==0][order(lnSqFeet[Air==0])],
col="blue"),
lines(sort(lnSqFeet[Air==1]),
fitted(model.2)[Air==1][order(lnSqFeet[Air==1])],
col="red")))
legend("topleft", col=c("blue", "red"), pch=1, lty=1, inset=0.02,
legend=c("No air conditioning", "Air conditioning"))

plot(x=fitted(model.2), y=residuals(model.2),
xlab="Fitted values", ylab="Residuals",
panel.last = abline(h=0, lty=2))

detach(realestate)

### Hospital infection risk (4-level categorical predictor, additive model)

• Load the infectionrisk data and select observations with Stay <= 14.
• Create indicator variables for regions.
• Fit a multiple linear regression model of InfctRsk on Stay + Xray + i2 + i3 + i4.
• Conduct an F-test to see if at least one of i2, i3, and i4 are useful (conclusion: the regression functions differ by region).
• Conduct an F-test to see if at least one of i2 and i3 are useful (conclusion: only the west region differs).
infectionrisk <- read.table("~/path-to-folder/infectionrisk.txt", header=T)
infectionrisk <- infectionrisk[infectionrisk\$Stay<=14,]
attach(infectionrisk)

i1 <- ifelse(Region==1,1,0) # NE
i2 <- ifelse(Region==2,1,0) # NC
i3 <- ifelse(Region==3,1,0) # S
i4 <- ifelse(Region==4,1,0) # W

model.1 <- lm(InfctRsk ~ Stay + Xray + i2 + i3 + i4)
summary(model.1)
#              Estimate Std. Error t value Pr(>|t|)
# (Intercept) -2.134259   0.877347  -2.433  0.01668 *
# Stay         0.505394   0.081455   6.205 1.11e-08 ***
# Xray         0.017587   0.005649   3.113  0.00238 **
# i2           0.171284   0.281475   0.609  0.54416
# i3           0.095461   0.288852   0.330  0.74169
# i4           1.057835   0.378077   2.798  0.00612 **
# ---
# Residual standard error: 1.036 on 105 degrees of freedom
# Multiple R-squared:  0.4198,  Adjusted R-squared:  0.3922
# F-statistic: 15.19 on 5 and 105 DF,  p-value: 3.243e-11

model.2 <- lm(InfctRsk ~ Stay + Xray)
anova(model.2, model.1)
#   Res.Df    RSS Df Sum of Sq      F  Pr(>F)
# 1    108 123.56
# 2    105 112.71  3    10.849 3.3687 0.02135 *

model.3 <- lm(InfctRsk ~ Stay + Xray + i4)
anova(model.3, model.1)
#   Res.Df    RSS Df Sum of Sq      F Pr(>F)
# 1    107 113.11
# 2    105 112.71  2   0.39949 0.1861 0.8305

detach(infectionrisk)

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