Use G*Power 3 to calculate power for modified model with alpha =.05 and NCP=6.3496 at d f=1, d f

Use G*Power 3 to calculate power for modified model with...

Key Concepts

Statistical Power

Statistical power is the probability that a test will correctly reject a false null hypothesis. It reflects the sensitivity of the test to detect an effect when one exists. Power depends on several factors such as sample size, effect size, significance level, and the structure of the model being tested. It is crucial for designing studies that are capable of detecting meaningful effects.

Degrees of Freedom

Degrees of freedom refer to the number of independent values or pieces of information that are available to estimate statistical parameters. In the context of model testing, increasing degrees of freedom generally means that more parameters are being estimated, which can affect the critical values of the test statistic. This, in turn, influences the test’s power, typically leading to lower power if other factors like the effect size or noncentrality parameter remain unchanged.

Noncentrality Parameter (NCP)

The noncentrality parameter is a measure used in power analysis that quantifies the degree of deviation of the test statistic from the null hypothesis distribution under the alternative hypothesis. It is directly related to the effect size and the sample size, and plays a key role in determining the power of a test. When the NCP is held constant, as degrees of freedom increase, the critical value increases, which often results in a reduction of power.

Model Complexity

Model complexity relates to the number of parameters or constraints within a statistical model, often represented by the degrees of freedom. As models become more complex (i.e., higher degrees of freedom), the test statistics are affected, which can make it harder to detect effects due to increased critical thresholds. This is why, with a fixed noncentrality parameter, increasing the degrees of freedom may lead to a decrease in statistical power.

Transcript

00:01 Hi guys in this problem we have a non -hypothesis h node such that p equals 0 .5 and h1 such that p does not equal 0 .5 okay so since the test is two -sided the statistics for a test where z of alpha over 2 so it's z over 0 .025 so it's 1 .96 okay okay so 1 minus beta, okay, the power, since the test is two -sided and the true population proportion, p equal 0 .6 and p note equals 0 .5, okay, the power of the test for n equals 100.

00:56 Okay, so the power, it's 1 ,000.

01:05 Peta okay so one minus here in peta can be calculated as phi of yeah just p node minus p plus z of alpha over 2 times the square root of p node times one minus p node over m okay over the square root of p node of p to over the square root of p node times 1 minus p over n okay so this is minus phi of p node minus p minus z of alpha over 2 times the square root of p node times 1 minus p node over n over the square root of p times 1 minus p over n okay so this is exactly as 1 minus q of negative 0 .04 negative q of negative 4 .04 okay so this is 0 .516 okay okay then for the same power for n equals 150 so the power it's uh equals 1 minus beta and substituting in the previous formula.

03:07 So it's 1 minus q of negative 0 .5 minus q of 4 .5.

03:18 Okay, so this is 0 .6915.

03:27 Okay, okay.

03:29 So for n equals 300.

03:32 So the power is 1 minus p.

03:37 And this is 1 minus q of negative 1 .54 minus q of negative 5 .54 okay so this is 0 .3 9382 so as we see if the sample size increases the power of the test increases okay okay then for part p since the test is two -sided so here we have z of alpha over 2 and we have alpha is 0 .01 so it's z of 0 .005 so it's 2 .58 okay okay so then for n equals 100 the power equal 1 minus beta so this is 1 minus beta so this is 1 minus q of 0 .59 minus q of negative 4 .67 okay so this is 0 .276 okay so then when n equals 150 so the power is 1 minus beta so it's 1 minus q of 0 .6 .6.

05:16 So, so it's 1 minus q of 0 .6.

05:19 Times q of negative 5 .13.

05:27 Okay, so this is 0 .448.

05:35 Okay, then when n equals 300, so the power equals 1 minus beta, so it's 1 minus 5 of negative 0 .9, minus 5 of negative 6 .17.

06:00 Okay, so this is 0 .8159...

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