Question

An experiment involving subjects with schizophrenia compared "personal therapy" to "family therapy." Only 2 out of 23 subjects assigned to the personal therapy group suffered psychotic relapses in the first year of the study, compared to 8 of the 24 subjects assigned to the family therapy group. \(^{29}\) Is this sufficient evidence to conclude, at the 0.05 level of significance, that the two types of therapies are not equally effective? Conduct Fisher's exact test using a nondirectional alternative.

Short answer

After conducting Fisher's exact test and comparing the P-value to the 0.05 significance level, determine if the null hypothesis should be rejected based on whether the P-value is less than or equal to 0.05.

Step-by-step solution

Set up the Contingency Table

Make a 2x2 contingency table with the data provided for the two different therapies and their outcomes, differentiating between those who suffered psychotic relapses and those who did not within the first year.

Calculate the Test Statistic using Fisher's Exact Test

Apply Fisher's exact test using the contingency table. Fisher's exact test is appropriate here because the sample sizes are small. Calculate the probability of the observed configuration of the contingency table, as well as the probability of all configurations that demonstrate an equal or greater level of association.

Determine the P-value

Sum the probabilities calculated in the previous step to find the P-value, which represents the probability of observing the given outcome or one more extreme if the null hypothesis is true.

Compare P-value with Significance Level

Compare the obtained P-value to the significance level of 0.05 to determine whether the evidence is sufficient to conclude that there is a significant difference between the effectiveness of personal therapy and family therapy.

Conclusion

If the P-value is less than or equal to 0.05, we reject the null hypothesis of equal effectiveness of the therapies. If it is greater, we do not have sufficient evidence to conclude a difference in effectiveness.

Key concepts

Contingency Table

When we conduct a study comparing outcomes across different groups, like the effectiveness of personal therapy versus family therapy in schizophrenia patients, we use a tool called a contingency table. In our exercise, this table was a 2x2 grid, reflecting the two types of therapy and whether patients suffered relapses or not. Here's why it's crucial:

• A contingency table organizes data into categories for comparison.
• It helps visualize the relationship between two categorical variables.
• It is the foundation for performing Fisher's exact test, especially when sample sizes are small.

Improving our understanding of contingency tables can help us better interpret statistical analyses and reinforce the concept of association between variables.

Statistical Significance

Determining whether an observed effect is real or just due to chance is where statistical significance comes into play. In the context of the exercise, we asked if personal therapy and family therapy have different effects on schizophrenia relapses within one year.

The statistical significance is established by a predetermined significance level, commonly set at 0.05. This means there's a 5% chance of concluding that an effect exists when it actually does not. When the computed P-value is below this cut-off, the results are said to be statistically significant, implying that it's unlikely the observed difference in therapy outcomes is due to chance alone. This threshold reinforces our confidence in the study's findings and guides our decision on the null hypothesis.

Null Hypothesis

In any experiment that seeks to explore the efficacy of different treatments, a starting assumption is made, known as the null hypothesis. It assumes there is no effect or difference between the groups being compared – in our case, that both personal therapy and family therapy are equally effective.

Fisher's exact test directly assesses the likelihood of obtaining the data we have if the null hypothesis were true. A P-value – derived from comparing the calculated probabilities of the observed and more extreme outcomes to the total possible outcomes – guides us. If this P-value is less than the significance level, in our exercise 0.05, we reject the null hypothesis, accepting the alternative that the effectiveness of treatments is indeed different. Proper understanding and handling of the null hypothesis are fundamental to interpreting the results of a Fisher's exact test and drawing meaningful conclusions from the data.