STM1001: Computer Lab 12 Solutions

Topic 11: Statistical Power and Sample Size Calculation

These are the solutions for Computer Lab 12, and use the G*Power software program (Faul et al. 2007), see also Faul et al. (2009).

1 G*Power

1.1 Overview

No answer required.

1.2 One Sample \(t\)-test

1.2.1

The required sample size value is \(n = 12\). You should obtain this result as shown in Figure 1.1 below.

Figure 1.1: One sample \(t\)-test G*Power output.

1.2.2

The new required sample size value is \(n=22\). You should obtain this result as shown in Figure 1.2 below.

Figure 1.2: One sample \(t\)-test G*Power output for new inputs.

1.2.3

There are several ways in which a sample size of \(30\) could be achieved.

Assume that you begin with the parameters specified in 1.2.1, i.e:

• The hypothesis test is two-tailed.
• The significance level is \(\alpha =0.05\).
• The null hypothesis value is \(5\).
• A mean difference of at least \(0.5\) is considered meaningful.
• The estimated standard deviation for the population is \(0.55\).
• We would like a sample size that ensures a power of at least \(0.8\).

If the mean difference considered meaningful is reduced, the sample size will increase - e.g. if this changes from \(0.5\) to \(0.295\), and all other parameters remain the same, a sample size result of \(n=30\) is obtained.

Other options include (either individually or in conjunction with other changes such as the change mentioned above):

• Decreasing the \(\alpha\) value
• Increasing the power value
• Having a larger estimated population standard deviation

1.3 Independent samples \(t\)-test

1.3.1

The required sample sizes are \(n_1 = 54\), and \(n_2 = 54\), for a total sample size of \(108\). You should obtain this result as shown in Figure 1.3 below.

Figure 1.3: Independent samples \(t\)-test G*Power output.

1.3.2

The new required sample sizes are \(n_1 = 11\), and \(n_2 = 11\), for a total sample size of \(22\). You should obtain this result as shown in Figure 1.4 below.

Figure 1.4: Independent samples \(t\)-test G*Power output for new inputs.

1.3.3

There are several ways in which a sample size of \(80\) could be achieved.

Assume that you begin with the parameters specified in 1.3.1, i.e:

• The hypothesis test is two-tailed.
• The significance level is \(\alpha =0.05\).
• The Group A mean is \(5\).
• A difference in mean between groups of at least \(0.3\) is considered meaningful.
• The estimated standard deviation for both groups is \(0.55\).
• We would like a sample size that ensures a power of at least \(0.8\).

In this instance, since we are aiming for a maximum sample size of 80 (i.e. less than the original 108), our focus is on adjusting the parameters so that we observe a decrease in the total sample size required.

Options which will lead to a decrease in the total sample size include:

• Increasing the \(\alpha\) value
• Decreasing the power value
• Increasing the value of the difference in means between the two groups that is considered meaningful
• Having groups with smaller estimated standard deviations

1.4 Paired \(t\)-test

1.4.1

The required sample size value is \(n = 50\). You should obtain this result as shown in Figure 1.5 below.

Figure 1.5: Paired \(t\)-test G*Power output.

1.4.2

The new required sample size value is \(n=20\). You should obtain this result as shown in Figure 1.6 below.

Figure 1.6: Paired \(t\)-test G*Power output for new inputs.

1.4.3

There are several ways in which a sample size of \(40\) could be achieved.

Assume that you begin with the parameters specified in 1.4.1, i.e:

• The hypothesis test is two-tailed.
• The significance level is \(\alpha =0.05\).
• The estimated mean starting weight is \(36.6\) kgs.
• An average change in weight between before and after weights of at least \(1.1\) kg is considered meaningful.
• The estimated standard deviations for the before and after weights are \(2.1\) and \(3.5\) respectively.
• The estimated correlation is \(0.63\).
• We would like a sample size that ensures a power of at least \(0.8\).

Options which will lead to a decrease in the total sample size include:

• Increasing the \(\alpha\) value
• Decreasing the power value
• Increasing the value of the minimum average change in weight between before and after weights that is considered meaningful
• Having a higher correlation between the two groups

1.5 Two-sample test of proportions

1.5.1

The required sample size values are \(n_1 = n_2 = 294\), for a total sample size of \(n=588\). You should obtain this result as shown in Figure 1.7 below.

Figure 1.7: Two-sample test of proportions G*Power result.

1.5.2

The new required sample size values are \(n_1 = n_2 = 127\), for a total sample size of \(n=254\). You should obtain this result as shown in Figure 1.8 below.

Figure 1.8: Two-sample test of proportions G*Power output for new inputs.

1.5.3

There are several ways in which a sample size of \(500\) could be achieved.

Assume that you begin with the parameters specified in 1.5.1, i.e:

• The hypothesis test is two-tailed.
• The significance level is \(\alpha =0.05\).
• The estimated percentage of interest under \(H_0\) is 70%.
• An observed difference of at least 10% is considered meaningful.
• We would like a sample size that ensures a power of at least \(0.8\).

Options which will lead to a decrease in the total sample size include:

• Increasing the \(\alpha\) value
• Decreasing the power value
• Increasing the difference between the two proportions that is considered meaningful (this can have a large impact)

2 Practice

Using a new power requirement of \(0.9\), we obtain the following results:

2.1 One Sample \(t\)-test

The required sample size value is now \(n = 15\). You should obtain this result as shown in Figure 2.1 below.

Figure 2.1: One sample \(t\)-test G*Power output, with power of \(0.9\).

2.2 Independent samples \(t\)-test

The required sample sizes are now \(n_1 = 72\), and \(n_2 = 72\), for a total sample size of \(144\). You should obtain this result as shown in Figure 2.2 below.

Figure 2.2: Independent samples \(t\)-test G*Power output, with power of \(0.9\).

2.3 Paired \(t\)-test

The required sample size value is now \(n = 67\). You should obtain this result as shown in Figure 2.3 below.

Figure 2.3: Paired \(t\)-test G*Power output.

2.4 Two-sample test of proportions

The required sample size values are now \(n_1 = n_2 = 392\), for a total sample size of \(n=784\). You should obtain this result as shown in Figure 2.4 below.

Figure 2.4: Two-sample test of proportions G*Power result, with power of \(0.9\).

That’s it! The final computer lab is done! If there were any parts you were unsure about, take a look back over the relevant sections of the Topic 11 material. Time to start revising for the exam…

References

Faul, F., E. Erdfelder, A. Buchner, and A. Lang. 2009. “Statistical Power Analyses Using G* Power 3.1: Tests for Correlation and Regression Analyses.” Behavior Research Methods 41 (4): 1149–60.

Faul, F., E. Erdfelder, A. Lang, and A. Buchner. 2007. ““G* Power 3: A Flexible Statistical Power Analysis Program for the Social, Behavioral, and Biomedical Sciences.” Behavior Research Methods 39 (2): 175–91.

These notes have been prepared by Rupert Kuveke. The copyright for the material in these notes resides with the author named above, with the Department of Mathematics and Statistics and with La Trobe University. Copyright in this work is vested in La Trobe University including all La Trobe University branding and naming. Unless otherwise stated, material within this work is licensed under a Creative Commons Attribution-Non Commercial-Non Derivatives License BY-NC-ND.