| Basic Commands | Probability Puzzles | Hypothesis Test, Count Data | Hypothesis Test, Measured Data | Confidence Interval, Count Data | Confidence Interval, Measured Data | Association / Correlation | Regression | Other Examples |

Birthweight-2

[paired sign test; see BIRTHWEIGHT-1 for a permutation test, Monte Carlo style, and BIRTHWEIGHT-3 for a paired permutation 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-2 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 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 nontreatment. The following procedure will avoid that problem. Calculate the direction of the birthweight changes for each woman with and without drug treatment. If birthweight after drug treatment is higher, score 1; if birthweight is lower or equal, score minus 1. (The question is whether there is any improvement, so a tie is no better than a decreased weight.) Because the recorded values are 11 with increases and 4 without, the sum is 7. This will be our benchmark statistic. Can a sum-of-signs as large as this arise by chance variations in the birthweights of babies born to 15 mothers? Specifically, if the chance of improvement is 50/50, are improvements likely to exceed with improvements of 7 or more?

1. For each of the 15 women, flip a coin. If you get heads, score 1, if tails, score minus 1.
2. Calculate the sum of these random fluctuations, 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

MAXSIZE scrboard 10000
REPEAT 9999
  NUMBERS (1 -1) sign

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

SAMPLE 15 sign sign$

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

SUM sign$ signsum$

calculate the mean of the simulated effects

SCORE signsum$ scrboard

and record these in "scrboard"

END
COUNT scrboard >= 7 more

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

DIVIDE more 9999 prob

convert to a proportion of the total trials

PRINT prob

Results

Frequency histogram of sum-of-signs

After 10,000 tests of 15 random throws of +1/-1, a sum-of-signs as large as 7 was reached 5.8% of the time.

Conclusion

We threw away all quantitative information and retained only whether the birthweight of an infant after drug treatment was higher or lower than that of an infant born previously to the same mother. This is a very conservative approach. In the simulation, a sum as large as 7 was reached about 5.8% of the time. We cannot conclude, therefore, that the infants born to mothers given drug treatment had higher birthweights.

References

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