{"id":1229,"date":"2017-01-09T18:52:43","date_gmt":"2017-01-09T23:52:43","guid":{"rendered":"https:\/\/sites.berry.edu\/vbissonnette\/?page_id=1229"},"modified":"2017-01-21T09:50:30","modified_gmt":"2017-01-21T14:50:30","slug":"illustrate-statistical-power","status":"publish","type":"page","link":"https:\/\/sites.berry.edu\/vbissonnette\/index\/stats-homework\/documentation\/illustrate-statistical-power\/","title":{"rendered":"Illustrate Statistical Power"},"content":{"rendered":"

To access this routine from within Stats Homework<\/em>, pull down the Probability<\/strong> menu, select Illustrate Areas in a Distribution<\/strong>, and then select Illustrate Statistical Power with the Normal Distribution<\/strong>.\u00a0 If you are using the stand-alone program, you will not need to do this.\u00a0 Here is the user interface:<\/p>\n

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This program models the situation you face when you are conducting a null-hypothesis test using\u00a0the normal distribution\u00a0(e.g., a one-sample z test of the mean).\u00a0 The distribution you see above is a model of your null hypothesis — i.e., you\u00a0start with the\u00a0assumption\u00a0that the effect size of your research is zero.\u00a0 The distribution you see here is a normal distribution with a mean of 100 and a standard deviation of 10.<\/p>\n

If you were to get a result that falls in one of the red tails -the areas of rejection- you would reject your null hypothesis and conclude that your result represents a\u00a0statistically-significant effect.\u00a0\u00a0The default is for you to assess a non-directional (i.e., two-tailed) hypothesis with an alpha = .05.\u00a0 So, the critical value of your z statistic would be +\/-\u00a01.96.\u00a0 This corresponds to a score of 119.60 in this distribution \u00a0([1.96 * 10] + 100 = 119.6).<\/p>\n

Ok, let’s consider an effect size greater than zero.\u00a0 Move the effect-size slider\u00a0back and forth, and then release it at\u00a010.0.\u00a0 Then, press the “Re-scale Plot” button:<\/p>\n

\"\"<\/a>Notice that a new distribution appears with a mean of 110 (this is 100 plus your effect size of 10).\u00a0 This is a model of an alternative\u00a0to your null — i.e., \u00a0that your result comes from a distribution with a mean\u00a0different from 100.\u00a0 If this were the result of your research, you would have obtained a z statistic value of 1.00: z = (research mean – null mean) \/\u00a0standard\u00a0deviation = (110 – 100) \/ 10 = 1.0.\u00a0 If this were an actual research result, this z value (1.0) would not fall in the area of rejection (i.e., a z value greater than 1.96), so you would fail to reject the null hypothesis.<\/p>\n

Imagine that you are randomly sampling results from this alternative distribution.\u00a0 Sometimes (17% of the time), the result you obtain would fall in the rejection area of the null distribution (this is the blue area highlighted in the graph).\u00a0 This area represents the statistical power of this test — the probability that you will reject the null hypothesis given that some alternative is actually true.\u00a0 This also tells us that 83% of the time, we will fail to reject the null.\u00a0 This is Beta — the probability that you fail to reject the null hypothesis when it is in fact false.\u00a0 Power is 1 – Beta.<\/p>\n

In the above situation, you are working with single scores (i.e., n = 1).\u00a0 In most research situations, you will be working with some sample with n greater than 1.\u00a0 Slide the “n” slider back and forth, and see how this changes the shape of the\u00a0distributions, and thus, the amount that they overlap.\u00a0 Notice that as n increases, the value of your z statistic increases, and the critical value required to reject your null hypothesis decreases.\u00a0 Slide this slider to 5 and release it.\u00a0 Then, press the “Re-Scale Plot” button:<\/p>\n

\"\"<\/a>Here you can see that the effect size\u00a0–the mean of the alternative distribution– is still 10.0 (1 standard deviation above the mean\u00a0in a normal distribution).\u00a0\u00a0 However, the standard error of the distributions is\u00a0now 4.47.\u00a0 Standard error = standard deviation \/ sqrt(n) = 10.0 \/ sqrt (5) = 4.47.\u00a0 Now, the z statistic is 2.24.\u00a0 z is equal to (alternative mean – null mean) \/ standard error =(110 – 100)\u00a0\/ (10.0 \/ sqrt(5)) = 2.24.\u00a0 If this were an actual research result this z statistic of 2.24 would fall in the area of rejection (a value greater than 1.96), which would lead you to reject the null hypothesis.<\/p>\n

Now, the area of the alternative distribution that falls in the\u00a0null’s rejection region is much greater — 61%.\u00a0\u00a0This illustrates one of the important dynamics of statistical power: given an effect size greater than zero, power will increase\u00a0as you increase sample size, making it more likely that you reject the null.<\/p>\n

Switch back and forth studying power (1 – Beta), and then the probability of making a Type II error (Beta).\u00a0 The option for this is found on the “Plot Options” menu.\u00a0 Also, switch back and forth between a two-tailed and a one-tailed hypothesis test.\u00a0\u00a0Side each of the sliders back and forth and study he effect that your action has on the blue area — the level of power in your research.\u00a0 You will learn about\u00a0how each of these factors\u00a0\u00a0affect the level of statistical power in your research:<\/p>\n