Caution: External email.
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,
10 North Greene Street
GRECC (BT/18/GR)
Baltimore, MD 21201-1524
Cell phone 443-418-5382
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