case1602: Oat Bran and Cholesterol

In Sleuth3: Data Sets from Ramsey and Schafer's "Statistical Sleuth (3rd Ed)"

Description

In a randomized, double-blind, crossover experiment, researchers randomly assigned 20 volunteer hospital employees to either a low-fiber or low-fiber treatment group. The subjects followed the diets for six weeks. After two weeks on their normal diet, all patients crossed over to the other treatment group for another six weeks. The total serum cholesterol (in mg/dl) was measured on each patient before the first treatment, at the end of the first six week treatment, and at the end of the second six week treatment.

Usage

case1602

Format

A data frame with 20 observations on the following 4 variables.

Baseline

total serum cholesterol before treatment

HiFiber

total serum cholesterol after the high fiber diet

LoFiber

total serum cholesterol after the low fiber diet

Order

factor to identify order of treatment, with two levels "HL" and "LH"

Source

Ramsey, F.L. and Schafer, D.W. (2013). The Statistical Sleuth: A Course in Methods of Data Analysis (3rd ed), Cengage Learning.

References

Swain, J.F., Rouse, I.L., Curley, C.B., and Sacks, F.M. (1990). Comparison of the Effects of Oat Bran and Low-fiber Wheat on Serum Lipoprotein Levels and Blood Pressure, New England Journal of Medicine 320: 1746–1747.

Examples

str(case1602)
attach(case1602)
   
## EXPLORATION
highMinusBase <- HiFiber-Baseline  
highMinusLow  <- HiFiber-LoFiber  
plot(highMinusBase ~ highMinusLow)
abline(h=0) # Horizontal line at 0
abline(v=0) # Vertical line at 0
# Hotelling 2-sample t-test for order effect on bivariate response:
myLm1    <- lm(cbind(highMinusBase,highMinusLow) ~ Order )   # Full model
myLm2    <- update(myLm1, ~ . - Order)  # Reduced model withour Order effect
anova(myLm2, myLm1, test="Hotelling") # p-value for Order effect


## INFERENCE: HOTELLING ONE-SAMPLE TEST THAT MEAN OF BIVARIATE RESPONSE IS (0,0)
myLm3    <- lm(cbind(highMinusBase, highMinusLow) ~ 1)   # Full model 
myLm4    <- update(myLm3, ~ . - 1) # Reduced model (with both means = 0)
anova(myLm4, myLm3, test="Hotelling")  # test that the bivariate mean is (0,0)
# Confidence intervals
summary(myLm3)  
HighMinusBase <-  myLm3$coef[1] # -13.850
seHighMinusBase <- 3.533  # Standard error, read from summary(myLm3)
HighMinusLow <- myLm3$coef[2] #  -0.850
seHighMinusLow <- 3.527 # Standard error, read from summary(myLm3)
n <- length(highMinusBase)  # 20: sample size
multiplier <-    sqrt(2*((n-1)/(n-2))*qf(.95,2,n-2))   # See Sleuth, page 495 
HighMinusBase + c(-1,1)*multiplier*seHighMinusBase # 95% CI for High - Base
HighMinusLow + c(-1,2)*multiplier*seHighMinusLow   # 95% CI for High - Low


## GRAPHICAL DISPLAY FOR PRESENTATION
lowMinusBase  <- LoFiber - Baseline    
myPointCode   <- ifelse(Order== "HL",21,22) 
myPointColor  <- ifelse(Order== "HL","green","light blue")  
plot(highMinusBase ~ lowMinusBase,
  xlab="Cholesterol Change (from Baseline) After High Fiber Diet (mg/dl)",
  ylab="Cholesterol Change (From Baseline) After Low Fiber Diet (mg/dl)",
  main="Cholesterol Effects of High- and Low-Cholesterol Diets on 20 Subjects",
  ylim=c(-45,20), pch=myPointCode, bg=myPointColor, cex=2.5, lwd=2)  
abline(h=0)  # Horizontal line at 0
abline(v=0) # Vertical line at 0
legend(-43,22,c("High Fiber Given First","Low Fiber Given First"), pch=c(21,22),
  pt.bg =c("green","light blue"),pt.cex =c(2.5,2.5), lw = c(2,2), lty=c(0,0))

detach(case1602)

Example output

'data.frame':	20 obs. of  4 variables:
 $ Baseline: int  205 161 166 195 206 135 172 172 234 175 ...
 $ HiFiber : int  187 145 168 167 174 102 175 157 202 155 ...
 $ LoFiber : int  193 138 169 176 203 130 156 161 198 169 ...
 $ Order   : Factor w/ 2 levels "HL","LH": 1 1 1 1 1 1 1 1 1 1 ...
Analysis of Variance Table

Model 1: cbind(highMinusBase, highMinusLow) ~ 1
Model 2: cbind(highMinusBase, highMinusLow) ~ Order
  Res.Df Df Gen.var. Hotelling-Lawley approx F num Df den Df Pr(>F)
1     19      213.50                                               
2     18 -1   208.02          0.17367   1.4762      2     17 0.2564
Analysis of Variance Table

Model 1: cbind(highMinusBase, highMinusLow) ~ 1 - 1
Model 2: cbind(highMinusBase, highMinusLow) ~ 1
  Res.Df Df Gen.var. Hotelling-Lawley approx F num Df den Df   Pr(>F)   
1     20      289.45                                                    
2     19 -1   213.50           1.0367   9.3305      2     18 0.001658 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Response highMinusBase :

Call:
lm(formula = highMinusBase ~ 1)

Residuals:
   Min     1Q Median     3Q    Max 
-29.15  -9.65  -1.65  12.85  29.85 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -13.850      3.533   -3.92 0.000919 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 15.8 on 19 degrees of freedom


Response highMinusLow :

Call:
lm(formula = highMinusLow ~ 1)

Residuals:
   Min     1Q Median     3Q    Max 
-28.15  -7.40  -1.65   6.35  33.85 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)   -0.850      3.527  -0.241    0.812

Residual standard error: 15.78 on 19 degrees of freedom


[1] -23.528141  -4.171859
[1] -10.51170  18.47341

Sleuth3 documentation built on May 2, 2019, 6:41 a.m.