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Birthweight-3

[paired permutation test; see BIRTHWEIGHT-1 for permutation test, Monte Carlo style. and BIRTHWEIGHT-2 for paired sign test]

Problem

A drug was administered to 15 women who had previously given birth to underweight babies, to determine whether the drug would increase the birthweight of the next infant. The results are shown in the table below. The top row shows birthweights of the 15 babies born after drug treatment; the second row shows birthweights of babies born before drug treatment; the bottom row shows the difference in birthweights of the first and second infants. Use the knowledge that these are paired observations to estimate whether the drug treatment significantly increased birthweight (adapted from Rosner, 1982, p. 257).

Birthweight-3 Table. Comparison of Birthweights of Babies Born to 15 Women Before and After Drug Treatment

Treatment condition	Birthweight of babies by mother
	#1	#2	#3	#4	#5	#6	#7	#8	#9	#10	#11	#12	#13	#14	#15	mean
After	6.9	7.6	7.3	7.6	6.8	7.2	8.0	5.5	5.8	7.3	8.2	6.9	6.8	5.7	8.6	7.08
Before	6.4	6.7	5.4	8.2	5.3	6.6	5.8	5.7	6.2	7.1	7.0	6.9	5.6	4.2	6.5	6.26

Note. Difference in means = .82

Null hypothesis (H₀): The treatment does not affect birthweight, and the difference in means between the two groups arises by chance. Alternative hypothesis (H₁): The treatment increases birthweight.

Resampling Procedure

In BIRTHWEIGHT-1 we tested whether two samples whose means differed by .82 pounds or more could have come from the same universe. Such a procedure allows variation from mother to mother (if it is substantial) to obscure possible variation between treatment and nontreatement. The following procedure will avoid that problem by permuting separately for each mother (rather than tossing all data together). The mean of the paired differences is .82. According to the null hypothesis that the treatment has no effect on birthweight, the difference in weight for each mother's babies could have gone in either direction. We will test to see whether a mean birthweight difference of .82 could have arisen by chance. An assumption here is that the stability of birthweights is related to the mother, that is, some mothers tend to have infants of the same weight, whereas others tend to have infants of considerably different weights. We ask, "Could the treatment group do this much better than the nontreatment group if the direction of the weight change is randomly assigned?"

1. For each of the 15 women, flip a coin. If you get heads, then take the observed difference in birthweights of babies born before and after the mothers had drug treatment. If the difference is negative, then multiply the weight difference by -1.
2. Calculate the sum of these randomized differences, and record that result.
3. Repeat (1) and (2) 999 times.
4. Find how often the simulation yields a sum equal to or greater than that found in the study.

Computer Implementation in Resampling Stats

DATA (6.9 7.6 7.3 7.6 6.8 7.2 8.0 5.5 5.8 7.3 8.2 6.9 6.8 5.7 8.6) treatmt

set up a vector with the data for the treated women

DATA (6.4 6.7 5.4 8.2 5.3 6.6 5.8 5.7 6.2 7.1 7.0 6.9 5.6 4.2 6.8) control

This vector contains the data from untreated women. The mean difference in birthweight is .82 (obtained with a hand calculator).

SUBTRACT treatmt control difs

the vector "difs" holds the values that we will randomly add or subtract

NUMBERS (1 -1) sign

generate a vector with only "1" and "-1" in it

REPEAT 999
  SAMPLE 15 sign sign$

generate a random list of "plus" and "minus"

MULTIPLY sign$ difs difs$

randomized direction of simulated weight changes

MEAN difs$ meandif$
  SCORE meandif$ scrboard

and record these in "scrboard"

END
COUNT scrboard >= .82 more

how often were the simulated effects greater than or equal to the observed effects?

DIVIDE more 999 prob 

convert to a proportion of the total trials

PRINT prob

Results

Frequency histogram of difference in mean birthweights

In several runs of size 999,

prob = 0.001

prob = 0.004

prob = 0.006

prob = 0.005

Conclusion

If drug treatment had no effect on the direction of birth weights, there is only about a 0.004 chance of obtaining a result like that observed. Hence the null hypothesis is rejected, and we conclude that infants born to mothers given drug treatment had higher birthweights. However, these data, by themselves, are not enough to show that the drug was the cause. Perhaps second-born infants tend to be heavier than first-born. This possibility can be tested by obtaining additional data for birthweights of infants born to women not given drug treatment.

References

Rosner, B. (1982). Fundamentals of biostatistics. Boston: Duxbury Press.