In a 5 -year follow-up of bilateral stimulation of the...

In a 5 -year follow-up of bilateral stimulation of the subthalamic nucleus among 49 patients with advanced Parkinson's disease, investigators assessed changes in symptoms as measured by the Unified Parkinson's Disease Rating Scale (range $=0$ to 108 , with higher values denoting more symptoms). Assume this measure follows a normal distribution. The mean score at baseline was $55.7 .$ The standard deviation of change from baseline to 5 years was 15.3 . 8.161 How much power does this study have to detect a mean difference of 5 units in the symptom scale if a twosided test is used with $\alpha=.05 ?$

In a 5 -year follow-up of bilateral stimulation of the subthalamic nucleus among 49 patients with advanced Parkinson's disease, investigators assessed changes in symptoms as measured by the Unified Parkinson's Disease Rating Scale (range $=0$ to 108 , with higher values denoting more symptoms). Assume this measure follows a normal distribution. The mean score at baseline was $55.7 .$ The standard deviation of change from baseline to 5 years was 15.3 .
8.161 How much power does this study have to detect a mean difference of 5 units in the symptom scale if a twosided test is used with $\alpha=.05 ?$

Key Concepts

Statistical Power

Statistical power is the probability that a statistical test will correctly reject a false null hypothesis. It reflects the test’s sensitivity to detect a true effect, such as a specified difference between groups, and is influenced by factors such as sample size, effect size, and the chosen significance level.

Hypothesis Testing

Hypothesis testing is the framework used to decide whether observed data deviate significantly from a null hypothesis. It involves formulating a null hypothesis (typically representing no effect or difference) and an alternative hypothesis (indicating the presence of an effect), then using statistical tests to determine if the evidence is strong enough to reject the null.

Significance Level (Alpha)

The significance level, often represented by alpha, defines the threshold for rejecting the null hypothesis in a statistical test. It sets the maximum probability of making a Type I error, which is rejecting a true null hypothesis. In many studies, including two-sided tests, a common alpha value used is 0.05.

Effect Size

Effect size is a quantitative measure of the magnitude of the difference or relationship being investigated, independent of sample size. It is often expressed as the difference in means relative to the standard deviation, making it a crucial element in power calculations as it helps determine the sensitivity of a study to detect meaningful differences.

Sample Size

Sample size is the number of observations or participants included in a study. It plays a pivotal role in the precision of estimated parameters and in determining the statistical power of a test. Larger sample sizes generally lead to higher power, making it easier to detect smaller effect sizes.

Normal Distribution Assumption

The normal distribution assumption posits that the variable of interest follows a bell-shaped curve, which is symmetric and defined by its mean and standard deviation. This assumption is important for many parametric tests, including those for comparing means, as it underpins the validity of the inference drawn from the test results.

Transcript

00:01 In this question, we have information from an australian study about the actn3 gene and three variants, rr, rx, and x, x.

00:10 We'll test whether the proportions of each variant are 0 .25, 0 .50, and 0 .25.

00:18 So let's start by looking at the sample size.

00:22 What we want to do is find our sample size in part a, and what we need to do for that is add up the three observed values 130 plus 226 plus 80 and that gives us a total of 436.

00:39 The next thing we want to do is write down the observed number of people with the rr variant and we can see that in our table up here in the rr column under the observed row we have 130.

00:53 Now let's find the expected number with the rr variant under the null hypothesis.

01:00 So you can see see i wrote the null hypothesis here that p1 equals 0 .25, p2 equals 0 .50, and p3 equals 0 .25, with the alternative hypothesis being at least one of those proportions is not correct.

01:17 So to find the expected number with the rr variant under h0, under the null hypothesis, what we need to do is calculate this.

01:28 Let's take 0 .25 .5.

01:31 Times the total, which was 436.

01:36 So 0 .25 times 436 is 109.

01:41 Now, x, x also had a proportion of 0 .25 under the null hypothesis.

01:49 So let's just fill that in at the same time, because we'll need that in a minute.

01:53 That's 109 as well.

01:55 And then rx, under the null hypothesis, had 0 .50 as well as the, the proportion.

02:03 So 0 .5 times 436 is 218.

02:09 Okay, so now what we have is our expected row completed as well, and we'll use that coming up in the next part.

02:17 So what is the expected number with the rr variant under the null hypothesis? 109.

02:25 So we can see that in the table, second row, first column.

02:30 So let's take a look then at the next question, which variant contributes most to the kai square statistic? so as we do that, we have to calculate the kai square statistic this way.

02:43 Remember that the kai squared statistic is going to be the sum of the squared difference observed minus expected divided by expected.

02:56 So the first term that we write will be from our first column rr, 130 minus 109 squared, divide by 109.

03:07 The next, let me move this up so you can see the column heading, rx.

03:12 The next will be 226 minus 218 quantity squared, divide by 218.

03:21 And then the third term will be 80, the observed value for xx, minus the expected count 109.

03:31 We'll square that and divide by 109.

03:35 So when we do this, it's always a good idea to write out the individual values next because we might want to know the contribution of each term.

03:46 And we do in this case.

03:47 The contribution of the first term from the first variant is 4 .05.

03:52 The next one works out to be 0 .29...

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