Thus, when sample size is increased, the power of the study is increased, and the probability of correctly rejecting the null hypothesis is increased. To increase power (the probability of rejecting . the null when the alternative is true) and maintain the same alpha level (the probability of incorrectly rejecting the null), you need to increase the sample size. In simpler terms, increasing power requires increasing sample size to maintain the same alpha level.
The problem is that increasing the sample size usually means increasing the cost of a study, so statisticians use other approaches to decide on an acceptable sample size. The first involves determining the desired effect size and then determining the sample size needed to achieve the corresponding power. Statisticians typically choose 0.80, or 80%, power and then decide whether the study is feasible given the required sample size.
I then perform a cost-benefit analysis on my final sample size to determine whether the study should be conducted. Another approach to study design is to determine what sample size is available and then what effect size is needed to achieve an adequate level of power (again, usually 0.80 or 80%). All of these ideas are related and you need to understand them to determine your sample size.
In other words, the sample size you need should give you enough power to correctly reject the null hypothesis.
S UMMARY
Your statistical test states that the p-value for the association between taking a tablet of drug A daily and lowering cholesterol is 0.02. You determine after the trial that the actual effect size from the medication was smaller than you first thought. If your study had an alpha of 0.05 and a power of 80%, calculate the chance that you made a type 2 error.
In each of the following cases, identify which type of error is possible: type one or type two. Your study examining the relationship between head trauma and grand mal seizures has an alpha of 0.10 and a beta of 0.80. The poorly designed pilot study examining the relationship between mold exposure and asthma shows a small effect size.
You recruit a large sample to try to enable your research team to successfully detect this effect size. If the study measures symptoms of neuropathy on a scale of 1 to 10, what level of measurement is this variable? What effect size is expected, and what does that mean in terms of the required sample size.
This study, examining the treatment of patients with drug A and neuropathy symptoms, uses an alpha of 0.05 and reports a p-value of 0.30. If this decision in Review Question 28 about the null hypothesis is wrong, what type of error might it be? If a researcher reports that the abuse of dextromethorphan is not related to age, but is wrong about this conclusion, what type of error is that.
What does this mean for the relationship between the p-value and the alpha in the study? Failing to reject the null; there is not enough evidence to show a link between drug A and cholesterol levels. A large effect size is expected, so only a small sample size should be needed to detect it at a statistically significant level.
C HI -S QUARE
IS THERE A DIFFERENCE?
C HI -S QUARE (X2) T EST
D EGREES OF F REEDOM
When you put your data into a statistical program, it will calculate the expected values for each of these cells, assuming the two variables are independent. You should then apply the chi-square test to see if the observed values are significantly different from the expected values at one degree of freedom. If the X2 result has a p-value that is significant (usually 0.05, depending on the alpha you use), then you reject the null hypothesis that the two variables are independent and conclude that there is an association between genders and postoperative transfusion.
Your study does not have the statistical power for you to say that the variables are not related. Also note that the chi-square test does not tell you the direction of the relationship or difference. For example, if you had a statistically significant X2 in the example about gender and postoperative transfusions, you can go back and see which gender had more transfusions.
The chi-square test is one of the simplest tests available in a subset of statistics called categorical data analysis. This test is now commonly used instead of Pearson's chi-square test when the sample size of a cell in the data is less than 5 (because Fisher's arguments with Pearson were ultimately proven correct). In any case, the two tests—Pearson's chi-square test and Fisher's exact test—are now very common statistics for use in clinical trials and scientific research.
Here is an encouraging example that is a bit more difficult than the one in the main text. The occupation of the husband has no effect on the marital happiness of women in this subgroup of occupations in the population from which we sampled. Now you know that there is a relationship between the husband's occupation and the wife's marital happiness (because your p-value is less than alpha, which means it is significant).
So you reject the null hypothesis that there is no correlation between a husband's occupation and a wife's marital happiness. Remember that the chi-square test only tells you that there is a better choice, not which one it is. Assuming your friend wants to get married in the first place, which proposal should she accept.
Second, the chi-square test is used to look for a statistically significant difference or relationship when there is a dependent or outcome variable at the nominal or ordinal level. If the result of the chi-square test has a p-value that is significant (less than 0.05 or whatever alpha you use), then you reject the null hypothesis. If the result of the chi-square test is not statistically significant (greater than 0.05 or the alpha of choice), then you cannot reject the null hypothesis.
Finally, the chi-square test does not tell you the direction of the relationship; only you can make that interpretation. If the entire school has 800 students and the ninth grade has 250 students, what percentage of the ninth grade population did you sample. If the entire school has a population of 800, what percentage of students are included in your sample.
Imagine you are the editor of the journal to which an article has been submitted for review using a chi-square test to determine whether boys or girls are more likely to participate in sports. After reviewing your screening tool, you realize that the following error was made: When the survey was administered to three of the men's sports teams, it was printed on only one side of the paper and should have been copied on both sides. As a result, half of the survey was missing when administered to these three teams.
What does this tell you about the validity of the screening tool in this situation? Knowing the results of the larger study should make the researcher question whether the conclusion of the smaller pilot study was what kind of error. Thirty-eight percent of the sample reported that coffee has an effect on their disease symptoms.
Nominal, mode, mode = participation in sports for men and for the total sample H0: There is no relationship between gender and sports participation. If your alpha is 0.05, yes, sports participation is significantly different for men and women. To determine whether patients with Crohn's disease and ulcerative colitis have different perceptions of the impact of coffee consumption on their disease process.
S TUDENT T -T EST
HOW CAN I FIND A DIFFERENCE IN THE TWO SAMPLE MEANS IF MY DEPENDENT VARIABLE IS AT THE INTERVAL
OR RATIO LEVEL?