Sample Size Issues for Conjoint Analysis

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2 58 Sample Size Issues for Conjoint Analysis Though most of the principles that influence sample size determination are based on statistics, successful researchers develop heuristics for quickly determining sample sizes based on experience, rules-of-thumb, and budget constraints. Let us begin our discussion by making a distinction between sampling and measurement error. Subsequent sections will discuss each of these sources of error. 7.1 Sampling Error versus Measurement Error Errors are deviations from truth. In marketing research we are always concerned with reducing error in cost-effective ways. Assuming that you have selected the appropriate modeling method, there are two main sources of error that cause preference data to deviate from truth. The first is sampling error. Sampling error occurs when samples of respondents deviate from the underlying population. If we have drawn a random sample (each population element has an equal probability of being selected), sampling error is due to chance. If, on the other hand, our sample is not random (for example, a convenience sample), the sampling errors may be systematic. With random sampling, we reduce sampling error by simply increasing the sample size. With nonrandom sampling, however, there is no guarantee that increasing sample size will make the samples more representative of the population. To illustrate sampling error, assume we wanted to figure out how far the average adult can throw a baseball. If we drew a random sample of thirty people, and by chance happened to include Ichiro Suzuki (outfielder for the Seattle Mariners), our estimate would likely be farther than the true distance for the average adult. It is important to note that the samples we use in marketing research are rarely random. Some respondents resist being interviewed and, by selecting themselves out of our study, are a source of nonresponse bias. A second source of error in conjoint data is measurement error. We reduce measurement error by having more or better data from each respondent. Consider again the example of the baseball toss. Suppose you are one of the study participants. You throw the ball, but you accidentally step into an uneven spot on the ground, and the ball does not go as far as you typically could throw it. If we asked you to take another few tosses, and averaged the results, we would reduce the measurement error and get a better idea of how far you could throw a baseball. In conjoint analysis, we reduce measurement error by including more conjoint questions. We recognize, however, that respondents get tired, and there is a limit beyond which we can no longer get reliable responses, and therefore a limit to the amount we can reduce measurement error.

3 7.2 Binary Variables and Proportions Binary Variables and Proportions Sampling error is expressed in terms of standard errors, confidence intervals, and margins of error. We can begin to understand what these terms mean by considering binary variables and proportions. In fact, we will spend a good deal of time talking about confidence intervals for proportions because the statistical principles can be applied to choice-based conjoint results and shares of choice in market simulations for all conjoint techniques. A binary variable is a categorical variable with exactly two levels, such as a yes/no item on a consumer survey or a true/false checklist item. Many product attributes in conjoint studies have exactly two levels. And consumer choice itself is binary to choose or not, to buy or not. Binary variables are usually coded as 1 for yes and 0 for no. Looking across a set of binary variables, we see a set of 1s and 0s. We can count the number of 1s, and we can compute the proportion of 1s, which is the number of 1s divided by the sample size n. In statistical theory, the sampling distribution of the proportion is obtained by taking repeated random samples from the population and computing the proportion for each sample. The standard error of the proportion is the standard deviation of these proportions across the repeated samples. The standard error of a proportion is given by the following formula: pq standard error of a proportion = (n 1) where p is the sample estimate of the proportion in the population, q = (1 p), and n is the sample size. Most of us are familiar with the practice of reporting the results of opinion polls. Typically, a report may say something like this: If the election were held today, Mike Jackson is projected to capture 50 percent of the vote. The survey was conducted by the XYZ company and has a margin of error of ±3 percent. What is margin of error? Margin of error refers to the upper and lower limits of a confidence interval. If we use what is known as the normal approximation to the binomial, we can obtain upper and lower limits of the 95% confidence interval for the proportion as pq margin of error for a proportion = ±1.96 (n 1) Going back to the polling report from XYZ company, we note that margin of error has a technical meaning in classical statistics. If XYZ were to repeat the poll a large number of times (with a different random sample each time), 95 percent of the confidence intervals associated with these samples would contain the true proportion in the population. But, of course, 5 percent of the confidence intervals would not contain the true proportion in the population. Confidence intervals are random intervals. Their upper and lower limits vary from one sample to the next.

4 60 Sample Size Issues for Conjoint Analysis Suppose we interview 500 respondents and ask whether they approve of the president s job performance, and suppose 65 percent say yes. What would be the margin of error of this statistic? We would compute the interval as follows: ±1.96 (0.65)(0.35) (500 1) = ±0.042 The margin of error is ±4.2 percent for a confidence interval from 60.8 to 69.2 percent. We expect 95 percent of the confidence intervals constructed in this way to contain the true value of the population proportion. Note that the standard error of the proportion varies with the size of the population proportion. So when there is agreement among people about a yes/no question on a survey, the value of p is closer to one or zero, and the standard error of the proportion is small. When there is disagreement, the value of p is closer to 0.50, and the standard error of the proportion is large. For any given sample size n, the largest value for the standard error occurs when p = When computing confidence intervals for proportions, then, the most conservative approach is to assume that the value of the population proportion is That is, for any given sample size and confidence interval type, p = 0.50 will provide the largest standard error and the widest margin of error. Binary variables and proportions have this special property for any given sample size n and confidence interval type, we know the maximum margin of error before we collect the data. The same cannot be said for continuous variables, which we discuss in the next section. 7.3 Continuous Variables and Means With continuous variables (ratings-based responses to conjoint profiles), one cannot estimate the standard error before fielding a study. The standard error of the mean is directly related to the standard deviation of the continuous variable, which differs from one study to the next and from one survey question to the next. Assuming a normal distribution, the standard error of the mean is given by standard error of the mean = standard deviation n And the margin of error associated with a 95% confidence interval for the mean is given by margin of error for the mean = ±1.96 ( standard error of the mean )

5 7.4 Small Populations and the Finite Population Correction 61 Suppose we had conducted an ACA study with forty respondent interviews. We want to estimate purchase likelihood for a client s planned product introduction with a margin of error of ±3 and a 95% confidence level. We run an ACA market simulation to estimate purchase likelihood on a 100-point scale, and the simulator reports the standard error next to the purchase likelihood estimate: Total Respondents = 40 Purchase Likelihood Standard Error Product A The margin of error is ± = ±6.00, so we need to cut the margin of error in half to achieve our ±3 target level of precision. We know that the standard error of the mean is equal to the standard deviation divided by the square-root of the sample size. To decrease the standard error by a factor of two, we must increase sample size by a factor of four. Therefore, we need to interview about 40 4 = 160 or 120 additional respondents to obtain a margin of error of ±3 for purchase likelihood. 7.4 Small Populations and the Finite Population Correction The examples we have presented thus far have assumed infinite or very large populations. But suppose that, instead of estimating the job performance rating of the president by the United States population at large, we wanted to estimate (with a margin of error of ±3 percent) the job performance rating of a school principal by members of the PTA. Suppose there are only 100 members of the PTA. How many PTA members do we need to interview to achieve a margin of error of ±3 percent for our estimate? First, we introduce a new term: finite population correction. The formula for the finite population correction is (N n) (N 1), where n is the sample size and N is the population size. The formula for the finite population correction is often simplified to (1 f), where f = n (N n) N, which is approximately equivalent to (N 1) for all except the smallest of populations. After a population reaches about 5,000 individuals, one can generally ignore the finite population correction factor because it has a very small impact on sample size decisions. Using the simplified finite population correction for a finite sample, the margin of error for a proportion and a 95% confidence interval is equal to ±1.96 pq (1 f) (n 1) The finite population correction may also be used for continuous variables and means.

10 66 Sample Size Issues for Conjoint Analysis A thorough discussion of sampling and measurement errors would require more time and many more pages. The reader is encouraged to consult other sources in these areas. For statistics and sampling see Snedecor and Cochran (1989) and Levy and Lemeshow (1999). For measurement theory see Nunnally (1967).

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