3 Loss of Power with Ordinal Data

An important factor which affects power but is often overlooked is the form of measurement used. So far we have considered only continuous, normally distributed variables, but of course, these are not always available in the biosocial sciences. An exhaustive treatment of the power of the ordinal classical twin study is beyond the scope of this text, but we shall simply illustrate the loss of power incurred when we use more crude scales of measurement. Consider the example above, but suppose this time that we wish to detect the presence of additive genetic effects, $a^2$, in the data. For the continuous case this is a trivial modification of the input file to fit a model with just $c$ and $e$ parameters. The chi-squared from running this program is $19.91$, and following the algebra above (equation 7.1) we see that we would require $2000\times 7.85/19.91 = 788$ pairs in total to be 80% certain of rejecting the hypothesis that additive genes do not affect variation when in the true world they account for 30%, with shared environment accounting for a further 20%. Suppose now that rather than measuring on a continuous scale, we have a dichotomous scale which bisects the population; for example, an item on which 50% say `yes' and 50% say no. The data for this case may be summarized as a contingency table, and we wish to generate tables that: (i) have a total sample size of 1000; (ii) reflect a correlation in liability of .5 for MZ and .35 for DZ twins; and (iii) reflect our threshold value of 0 to give 50% either side of the threshold. Any routine that will compute the bivariate normal integral for given thresholds and correlation is suitable to generate the expected proportions in each cell. In this case we use a short Mx script (Neale, 1991) to generate the data for PRELIS. We can use the weight variable option in PRELIS to indicate the cell counts for our contingency tables. Thus, the PRELIS script might be:

Power calculation MZ twins
DA NI=3 NO=0 
LA; SIM1 SIM2 FREQ
RA FI=expectmz.frq
WE FREQ
OR sim1 sim2
OU MA=PM SM=SIMMZ.COV SA=SIMMZ.ASY PA

with the file expectmz.frq looking like this:

0 0 333.333
0 1 166.667
1 0 166.667
1 1 333.333

A similar approach with the DZ correlation and thresholds gives expected frequencies which can be used to compute the asymptotic variance of the tetrachoric correlation for this second group. The simulated DZ frequency data might appear as

0 0 306.9092
0 1 193.0908
1 0 193.0908
1 1 306.9092

The cells display considerable symmetry -- there are as many concordant `no' pairs as there are concordant `yes' pairs because the threshold is at zero. Running PRELIS generates output files, and we can see immediately that the correlations for MZ and DZ twins remain the desired .5 and .35 assumed in the population. The next step is to feed the correlation matrix and the weight matrix (which only contains one element, the asymptotic variance of the correlation between twins) into Mx, in place of the covariance matrix that we supplied for the continuous case. This can be achieved by changing just three lines in each group of our Mx power script:

Data NGroups=2 NInput_vars=2 NObservations=1000
PMatrix File=SIMMZ.COV
ACov File=SIMMZ.ASY

with corresponding filenames for the DZ group, of course. When we fit the model to these summary statistics we observe a much smaller $\chi^2$ than we did for the continuous case; the $\chi^2$ is only 6.08, which corresponds to a requirement of 2,582 pairs in total for 80% power at the .05 level. That is, we need more than three times as many pairs to get the same information about a binary item than we need for a continuous variable. The situation further deteriorates as we move the threshold to one side of the distribution. Simulating contingency tables, computing tetrachorics and weight matrices, and fitting the false model when the threshold is one standard deviation (SD) to the right (giving 15.9% in one category and 84.1% in the other), the $\chi^2$ is a mere 3.29, corresponding a total sample size of 4,772 total pairs. More extreme thresholds further reduce power, so that for an item (or a disease) with a 95:5% split we would require 13,534 total pairs. Only in the largest studies could such sample sizes be attained, and they are quite unrealistic for data that could be collected by personal interview or laboratory measurement. On the positive side, it seems unlikely that given the advantages of the clinical interview or laboratory setting, our only measure could be made at the crude `yes or no' binary response level. If we are able to order our data into more than two categories, some of the lost power can be regained. Following the procedure outlined above, and assuming that there are two thresholds, one at $-1$ SD and one at $+1$ SD, then the $\chi^2$ obtained is 8.16, corresponding to `only' 1,924 pairs for 80% chance of finding additive variance significant at the .05 level. If one threshold is 0 and the other at 1 SD then this rises slightly to a $\chi^2$ of 9.07, or 1,730 pairs. Further improvement can be made if we increase the measurements to comprise four categories. For example, with thresholds at $-1$, $0$, and $1$ SD the $\chi^2$ is 12.46, corresponding to a sample size of 1,240 twin pairs.

While estimating tetrachoric correlations from a random sample of the population has considerable advantages, it is not always the method of choice for studies focused on a single outcome, such as schizophrenia. In cases where the base rates are so low (e.g., 1%) then it becomes inefficient to sample randomly, and an ascertainment scheme in which we select cases and examine their relatives is a practical and powerful alternative, if we have good information on the base rate in the population studied. The necessary power calculations can be performed using the computer packages LISCOMP (Muthén, 1987) or Mx (Neale, 1997).