[R] Difference in p-value obtained in an interrupted time-series analysis (containing main effects and and interaction) vs. a simple regression containing only main effects.

[Sorkin, John]

The question that follows in NOT an R question, but rather a statistics question. I hope you will forgive my statistics question.

I am investigating interrupted time-series analysis. My data has two periods, period 0 and period 1. In period 0 the slope is positive. In period 2 the slope is negative:

mydata<- structure(list(period = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
                                   0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), 
                        time2 = c(1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 
                                  13L, 14L, 15L, 16L, 1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 
                                  11L, 12L, 13L, 14L, 15L, 16L), time = 1:32, 
                        y = c(0.0190981041979942, 
                              -0.648843854645016, 2.62941433593151, 2.97383527060948, 2.68476280937483, 
                              2.51909810419799, 1.85115614535498, 5.12941433593151, 5.47383527060948, 
                              5.18476280937483, 5.01909810419799, 4.35115614535498, 7.62941433593151, 
                              7.97383527060948, 7.68476280937483, 7.51909810419799, 6.35115614535498, 
                              8.12941433593151, 6.97383527060948, 5.18476280937483, 3.51909810419799, 
                              1.35115614535498, 3.12941433593151, 1.97383527060948, 0.184762809374828, 
                              -1.48090189580201, -3.64884385464502, -1.87058566406849, 
                              -3.02616472939052, -4.81523719062517, -6.48090189580201, 
                              -8.64884385464502)), row.names = c(NA, -32L), class = "data.frame")
mydata
plot(mydata$time,mydata$y)
title("Data Used in my analyses")

# Regression of y on time, period 0 (16-data points)
fit0 <- lm(y~time,data=mydata[1:16,])
summary(fit0) #(16-data points)
abline(fit0,col="green")
cat("Regression period 0:", "beta=",summary(fit0)$coefficients["time","Estimate"],"p-value=",summary(fit0)$coefficients["time","Pr(>|t|)"],"\n")

# Regression of y on time, period 1 (16-data points)
fit1 <- lm(y~time,data=mydata[17:32,]) #(16-data points)
abline(fit1,col="red")
summary(fit1)
cat("Regression period 1:", "beta=",summary(fit1)$coefficients["time","Estimate"],"p-value=",summary(fit1)$coefficients["time","Pr(>|t|)"],"\n")

# Regression of y on time, period 0 and 1 (32-data points)
fit2 <- lm(y~time+period+time*period,data=mydata[1:32,])
summary(fit2) #(32-data points)
cat("Regression period 1 and 2 using interaction:", "beta=",summary(fit2)$coefficients["time","Estimate"],"p-value=",summary(fit2)$coefficients["time","Pr(>|t|)"],"\n")


Please note that the regression of y on time in period 0 (fit 0) returns
time slope=0.52358 p=2.86e-07

Please note that the regression of y on time in period 0 (fit 2) and 1 returns
time slope=0.52358 p=1.50e-09

The time slope are the same in the two models, fit0 and fit2, however the p-values are different, 2.86e-07 (fit 0) vs. 1.50e-09 (fit 2) a two-orders of magnitude improvement in the p-value!
I am not surprised that the time slopes are the same.  I am shocked that the p-values are different. While fit 0 used 16 lines of data, and fit 2 used 32 lines of data, the number of values that were used to compute the time slopes in period 0 by the two models are the same, 16. This being the case, why are the p-values of the time slopes different in fit0 vs. fit2?

Thank you,
John
 
P.S. This question is NOT homework. I am many years beyond being a student (at least a student in a class), but I am a teacher of a class (and a life-long student). The question comes from a discussion I had with a student in one of my classes.  





John David Sorkin M.D., Ph.D.
Professor of Medicine, University of Maryland School of Medicine;
Associate Director for Biostatistics and Informatics, Baltimore VA Medical Center Geriatrics Research, Education, and Clinical Center; 
PI Biostatistics and Informatics Core, University of Maryland School of Medicine Claude D. Pepper Older Americans Independence Center;
Senior Statistician University of Maryland Center for Vascular Research;

Division of Gerontology and Paliative Care,
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GRECC (BT/18/GR)
Baltimore, MD 21201-1524
Cell phone 443-418-5382