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Tumors

Problem

In a study by the US Food and Drug Administration, rats were treated with four dosages of a carcinogen (see Cytel Software Corporation, n.d.). The number of rats that developed tumors was determined over the next two years, as is shown in the table below. One way to analyze this data is with the stratified Cochran-Armitage trend test, in which the sum of the products of the number of tumors multiplied by dosage is calculated (Cochran-Armitage = sum of (dosage x #tumors). In the case of these data, the result is 207. Is this observed value - 207 - high in comparison to what chance might produce?

Tumors Table. Numbers of Tumors Developed by Rats Given Doses of a Carcinogen

	# Rats	# Doses	# Tumors
Group 1	48	0	1
Group 2	47	1	2
Group 3	48	5	1
Group 4	46	50	4
Totals	189	--	8

Note. Data are adapted from Cytel Software Corporation, n.d.

Note that the Cochran-Armitage statistic measures the extent of association between one variable and another. It is at its highest when there is perfect direct correlation. You can test this by using the above data and comparing the dosage vector {0 1 5 50} against various orderings of the tumors vector {1 2 1 4}. You will find that the sum of the products -- the Cochran-Armitage statistic -- is at its highest when the latter vector is arranged from lowest to highest {1 1 2 4}.

Null hypothesis (H₀): There is no association between carcinogen dosage and number of tumors. Alternative hypothesis (H₁): There is a positive association between carcinogen dosage and number of tumors.

Resampling Procedure

Overall, 8 rats developed tumors, and 181 did not. We can simulate a chance association between dosage and number of tumors by randomly drawing (without replacement) sets of 48 (to correspond to the 0-dose group), 47 (to correspond to the 1-dose group), 48 (to correspond to the 5-dose group), and 46 (to correspond to the 50-dose group). Then we recalculate the Cochran-Armitage statistic and observe whether this resampled value is as high as the observed value (which is at its highest when there is perfect positive association).

1. Take 189 pieces of paper to represent the 189 rats used. Mark 8 of these pieces with "T," to signify the number of rats that developed tumors.
2. Shuffle the pieces and take 48 of them to simulate the rats in group 1 given no carcinogen.
3. Take without replacement 47 pieces to simulate the rats in group 2 (1 dose). Count the number of papers marked "T" in this group.
4. Take 48 papers for group 3 (5 doses). Count the number of papers marked "T," and multiply this value by 5.
5. Take the remaining 46 papers (for group 4, 50 doses), count the number that were marked, and multiply this value by 50.
6. Add up all the results of (3), (4), and (5). Record the result.
7. Repeat (2) through (6) about 10,000 times. Determine how often the randomized simulation equals or exceeds the observed value of 207.

Computer Implementation in Resampling Stats

MAXsize scrboard 10000

we have to allow extra space for many repetitions of the simulation

COPY (1 2 1 4) tumors
COPY (0 1 5 50) doses

first, we let Resampling Stats calculate the actual value of the test statistic

MULTIPLY tumors doses products
SUM products ca-stat
PRINT ca-stat

this is the real-world value; now we begin the simulation

URN 8#1 181#0 rats

a small number of rats (8 of 189) developed tumors

REPEAT 9999
  SHUFFLE rats rats$

shuffle the original data into a randomized simulated universe

TAKE rats$ 1,48 group1

There were 48, 46, 47, and 45 rats given 0, 1, 5, and 50 doses of a carcinogen, respectively. Simulate a group of 48 rats given 0 doses

TAKE rats$ 49,95 group2

similarly, simulate 46 rats given 1 dose, and so on for the remaining two groups

TAKE rats$ 96,143 group3
  TAKE rats$ 144,189 group4
  COUNT group1 =1 group1$

how many tumors developed in each group?

COUNT group2 =1 group2$
  COUNT group3 =1 group3$
  COUNT group4 =1 group4$
  CONCAT group1$ group2$ group3$ group4$ allgroup

we concatenate results so we can multiply two vectors together in one step

MULTIPLY doses allgroup prod$

Voila! Each element of "doses" is multiplied by corresponding element of "allgroup" and the results stored in "prod$."

SUM prod$ sumprod$

add the simulated dose-response products together

SCORE sumprod$ scrboard
END
HISTOGRAM scrboard
COUNT scrboard >=207 more

how often did the simulation produce a result at least as great as the observed statistic?

DIVIDE more 9999 prob

divide by the number of repeats to obtain the answer as a proportion

PRINT prob

Results

Frequency histogram of resampled Cochran-Armitage statistic

ca-stat = 207

prob = 0.070607

Conclusion

The observed statistic of 207 was exceeded in 7% of the simulations. Therefore, the null hypothesis cannot be rejected. This experiment does not provide evidence that the tested carcinogen caused tumors in rats. The distribution of the simulated statistic is extremely lumpy. This results from a formula where the number of tumors in the 50-dose group (group 4) is responsible for almost all the final statistic, causing the distribution to peak at multiples of 50.

References

Cytel Software Corporation. (n.d.) FDA animal toxicology data yields sky-scraper distribution for stratified trend test (StatXact Fact Sheet). (Available from Cytel Software Corporation, 137 Erie St., Cambridge, MA 02139)