Tests for Two Correlations

Transcription

1 PASS Sample Sze Software Chapter 805 Tests for Two Correlatons Introducton The correlaton coeffcent (or correlaton), ρ, s a popular parameter for descrbng the strength of the assocaton between two varables. The correlaton coeffcent s the slope of the regresson lne between two varables when both varables have been standardzed. It ranges between plus and mnus one. Ths chapter covers the case n whch you want to test the dfference between two correlatons, each comng from a separate sample. Snce the correlaton s the standardzed slope between two varables, you could also apply ths procedure to the case n whch you want to test whether the slopes n two groups are equal. Test Procedure In the followng dscusson, ρ s the populaton correlaton coeffcent and r s the value calculated from a sample. The testng procedure s as follows. H 0 s the null hypothess that ρ1 = ρ2. H A represents the alternatve hypothess that ρ1 ρ2 (one-taled hypotheses are also avalable). To construct the hypothess test, transform the correlatons usng the Fsher-z transformaton. z = Z = 1 1+ r log 2 1 r 1 1+ ρ log 2 1 ρ Ths transformaton s used because the combned dstrbuton of r 1 and r 2 s too dffcult to work wth, but the dstrbutons of z 1 and z 2 are approxmately normal. Note that the reverse transformaton s r = e e Once the correlatons have been converted nto z values, the normal dstrbuton may be used to conduct the test of Z Z. The standard devaton of the dfference s gven by 1 2 z z + e e z z σ z z = N 3 + N NCSS, LLC. All Rghts Reserved.

2 PASS Sample Sze Software Tests for Two Correlatons The test statstc s gven by z = ( z z ) ( Z Z ) σ z1 z2 Note that the lower case z s represent the values calculated from the two samples and the upper case Z s represent the hypotheszed populaton values. Calculatng the Power 1. Fnd z α such that 1 Φ( z α ) = α, where Φ( x) s the area under the standardzed normal curve to the left of x. 2. Calculate: Z = 1 3. Calculate: Z = ρ 1 log 2 1 ρ ρ 2 log 2 1 ρ 2 4. Calculate: σ z z = N 3 + N Calculate: x = Z Z + zασ 6. Calculate: z = x a 1 2 z1 z2 a σ a z1 z2 7. Calculate: Power = ( ) 1 Φ z a Procedure Optons Ths secton descrbes the optons that are specfc to ths procedure. These are located on the Desgn tab. For more nformaton about the optons of other tabs, go to the Procedure Wndow chapter. Desgn Tab The Desgn tab contans most of the parameters and optons that you wll be concerned wth. Solve For Solve For Ths opton specfes the parameter to be solved for from the other parameters. Under most stuatons, you wll select ether Power or Sample Sze (N1). Select Sample Sze (N1) when you want to calculate the sample sze needed to acheve a gven power and alpha level. Select Power when you want to calculate the power of an experment NCSS, LLC. All Rghts Reserved.

3 PASS Sample Sze Software Test Tests for Two Correlatons Alternatve Hypothess Ths opton specfes the alternatve hypothess. Ths mplctly specfes the drecton of the hypothess test. The null hypothess s always H0: ρ1 = ρ2. Possble selectons are: Ha: ρ1 ρ2 Ths s the most common selecton. It yelds the two-taled test. Use ths opton when you are testng whether the correlaton values are dfferent, but you do not want to specfy beforehand whch value s larger. Ha: ρ1 < ρ2 Ths opton yelds a one-taled test. When you use ths opton, you should be careful to enter values for ρ1 and ρ2 that follow ths relatonshp. Ha: ρ1 > ρ2 Ths opton yelds a one-taled test. When you use ths opton, you should be careful to enter values for ρ1 and ρ2 that follow ths relatonshp. Power and Alpha Power Ths opton specfes one or more values for power. Power s the probablty of rejectng a false null hypothess, and s equal to one mnus Beta. Beta s the probablty of a type-ii error, whch occurs when a false null hypothess s not rejected. In ths procedure, a type-ii error occurs when you fal to reject the null hypothess of equal correlatons when n fact they are dfferent. Values must be between zero and one. Hstorcally, the value of 0.80 (Beta = 0.20) was used for power. Now, 0.90 (Beta = 0.10) s also commonly used. A sngle value may be entered here or a range of values such as 0.8 to 0.95 by 0.05 may be entered. Alpha Ths opton specfes one or more values for the probablty of a type-i error. A type-i error occurs when you reject the null hypothess of equal correlatons when n fact they are equal. Values of alpha must be between zero and one. Hstorcally, the value of 0.05 has been used for alpha. Ths means that about one test n twenty wll falsely reject the null hypothess. You should pck a value for alpha that represents the rsk of a type-i error you are wllng to take n your expermental stuaton. You may enter a range of values such as or 0.01 to 0.10 by Sample Sze (When Solvng for Sample Sze) Group Allocaton Select the opton that descrbes the constrants on N1 or N2 or both. The optons are Equal (N1 = N2) Ths selecton s used when you wsh to have equal sample szes n each group. Snce you are solvng for both sample szes at once, no addtonal sample sze parameters need to be entered NCSS, LLC. All Rghts Reserved.

4 PASS Sample Sze Software Tests for Two Correlatons Enter N1, solve for N2 Select ths opton when you wsh to fx N1 at some value (or values), and then solve only for N2. Please note that for some values of N1, there may not be a value of N2 that s large enough to obtan the desred power. Enter N2, solve for N1 Select ths opton when you wsh to fx N2 at some value (or values), and then solve only for N1. Please note that for some values of N2, there may not be a value of N1 that s large enough to obtan the desred power. Enter R = N2/N1, solve for N1 and N2 For ths choce, you set a value for the rato of N2 to N1, and then PASS determnes the needed N1 and N2, wth ths rato, to obtan the desred power. An equvalent representaton of the rato, R, s N2 = R * N1. Enter percentage n Group 1, solve for N1 and N2 For ths choce, you set a value for the percentage of the total sample sze that s n Group 1, and then PASS determnes the needed N1 and N2 wth ths percentage to obtan the desred power. N1 (Sample Sze, Group 1) Ths opton s dsplayed f Group Allocaton = Enter N1, solve for N2 N1 s the number of tems or ndvduals sampled from the Group 1 populaton. N1 must be 2. You can enter a sngle value or a seres of values. N2 (Sample Sze, Group 2) Ths opton s dsplayed f Group Allocaton = Enter N2, solve for N1 N2 s the number of tems or ndvduals sampled from the Group 2 populaton. N2 must be 2. You can enter a sngle value or a seres of values. R (Group Sample Sze Rato) Ths opton s dsplayed only f Group Allocaton = Enter R = N2/N1, solve for N1 and N2. R s the rato of N2 to N1. That s, R = N2 / N1. Use ths value to fx the rato of N2 to N1 whle solvng for N1 and N2. Only sample sze combnatons wth ths rato are consdered. N2 s related to N1 by the formula: where the value [Y] s the next nteger Y. N2 = [R N1], For example, settng R = 2.0 results n a Group 2 sample sze that s double the sample sze n Group 1 (e.g., N1 = 10 and N2 = 20, or N1 = 50 and N2 = 100). R must be greater than 0. If R < 1, then N2 wll be less than N1; f R > 1, then N2 wll be greater than N1. You can enter a sngle or a seres of values NCSS, LLC. All Rghts Reserved.

5 PASS Sample Sze Software Tests for Two Correlatons Percent n Group 1 Ths opton s dsplayed only f Group Allocaton = Enter percentage n Group 1, solve for N1 and N2. Use ths value to fx the percentage of the total sample sze allocated to Group 1 whle solvng for N1 and N2. Only sample sze combnatons wth ths Group 1 percentage are consdered. Small varatons from the specfed percentage may occur due to the dscrete nature of sample szes. The Percent n Group 1 must be greater than 0 and less than 100. You can enter a sngle or a seres of values. Sample Sze (When Not Solvng for Sample Sze) Group Allocaton Select the opton that descrbes how ndvduals n the study wll be allocated to Group 1 and to Group 2. The optons are Equal (N1 = N2) Ths selecton s used when you wsh to have equal sample szes n each group. A sngle per group sample sze wll be entered. Enter N1 and N2 ndvdually Ths choce permts you to enter dfferent values for N1 and N2. Enter N1 and R, where N2 = R * N1 Choose ths opton to specfy a value (or values) for N1, and obtan N2 as a rato (multple) of N1. Enter total sample sze and percentage n Group 1 Choose ths opton to specfy a value (or values) for the total sample sze (N), obtan N1 as a percentage of N, and then N2 as N - N1. Sample Sze Per Group Ths opton s dsplayed only f Group Allocaton = Equal (N1 = N2). The Sample Sze Per Group s the number of tems or ndvduals sampled from each of the Group 1 and Group 2 populatons. Snce the sample szes are the same n each group, ths value s the value for N1, and also the value for N2. The Sample Sze Per Group must be 2. You can enter a sngle value or a seres of values. N1 (Sample Sze, Group 1) Ths opton s dsplayed f Group Allocaton = Enter N1 and N2 ndvdually or Enter N1 and R, where N2 = R * N1. N1 s the number of tems or ndvduals sampled from the Group 1 populaton. N1 must be 2. You can enter a sngle value or a seres of values. N2 (Sample Sze, Group 2) Ths opton s dsplayed only f Group Allocaton = Enter N1 and N2 ndvdually. N2 s the number of tems or ndvduals sampled from the Group 2 populaton. N2 must be 2. You can enter a sngle value or a seres of values NCSS, LLC. All Rghts Reserved.

6 PASS Sample Sze Software Tests for Two Correlatons R (Group Sample Sze Rato) Ths opton s dsplayed only f Group Allocaton = Enter N1 and R, where N2 = R * N1. R s the rato of N2 to N1. That s, R = N2/N1 Use ths value to obtan N2 as a multple (or proporton) of N1. N2 s calculated from N1 usng the formula: where the value [Y] s the next nteger Y. N2=[R x N1], For example, settng R = 2.0 results n a Group 2 sample sze that s double the sample sze n Group 1. R must be greater than 0. If R < 1, then N2 wll be less than N1; f R > 1, then N2 wll be greater than N1. You can enter a sngle value or a seres of values. Total Sample Sze (N) Ths opton s dsplayed only f Group Allocaton = Enter total sample sze and percentage n Group 1. Ths s the total sample sze, or the sum of the two group sample szes. Ths value, along wth the percentage of the total sample sze n Group 1, mplctly defnes N1 and N2. The total sample sze must be greater than one, but practcally, must be greater than 3, snce each group sample sze needs to be at least 2. You can enter a sngle value or a seres of values. Percent n Group 1 Ths opton s dsplayed only f Group Allocaton = Enter total sample sze and percentage n Group 1. Ths value fxes the percentage of the total sample sze allocated to Group 1. Small varatons from the specfed percentage may occur due to the dscrete nature of sample szes. The Percent n Group 1 must be greater than 0 and less than 100. You can enter a sngle value or a seres of values. Effect Sze ρ1 (Correlaton Group 1) Specfy the value of the populaton correlaton coeffcent of group one. Possble values range between plus and mnus one. You can enter a sngle value or a range of values separated by commas or blanks. Note that the power depends on the specfc values of ρ 1 and ρ 2, not just ther dfference. Hence, ρ 1 = 0 and ρ 2 = 0.3 wll have a dfferent power from ρ 1 = 0.3 and ρ 2 = 0.6. ρ2 (Correlaton Group 2) Specfy the value of the populaton correlaton coeffcent from group two under the alternatve hypothess. Possble values range between plus and mnus one. You can enter a sngle value or a range of values separated by commas or blanks NCSS, LLC. All Rghts Reserved.

7 PASS Sample Sze Software Tests for Two Correlatons Example 1 Fndng the Power A researcher wants to compare the relatonshp between weght and heart rate n males and females. If the correlaton between weght and heart rate s 0.3 n a sample of 100 males and 0.5 n a sample of 100 females, what s the power of a two sded test for the dfference between correlatons at the 0.01 and 0.05 sgnfcance levels? Also compute the power for samples of 20, 200, 300, 400, and 600. Setup Ths secton presents the values of each of the parameters needed to run ths example. Frst, from the PASS Home wndow, load the Tests for Two Correlatons procedure wndow by expandng Correlaton, then Correlaton, then clckng on Test (Inequalty), and then clckng on Tests for Two Correlatons. You may then make the approprate entres as lsted below, or open Example 1 by gong to the Fle menu and choosng Open Example Template. Opton Value Desgn Tab Solve For... Power Alternatve Hypothess... Ha: ρ1 ρ2 Alpha Group Allocaton... Equal (N1 = N2) Sample Sze Per Group ρ1 (Correlaton Group 1) ρ2 (Correlaton Group 2) Annotated Output Clck the Calculate button to perform the calculatons and generate the followng output. Numerc Results Numerc Results when Ha: ρ1 ρ2 Power N1 N2 N ρ1 ρ2 ρ1 - ρ2 Alpha Report Defntons Power s the probablty of rejectng a false null hypothess. N1 and N2 are the number of tems sampled from each populaton. N s the total sample sze, N1 + N2. ρ1 s the value of both correlatons under the null hypothess. ρ2 s the correlaton n group two under the alternatve hypothess. ρ1 - ρ2 s the dfference between populaton correlatons at whch power and sample sze calculatons are made. Alpha s the probablty of rejectng a true null hypothess NCSS, LLC. All Rghts Reserved.

8 PASS Sample Sze Software Tests for Two Correlatons Summary Statements Group sample szes of 20 and 20 acheve 3% power to detect a dfference of between the null hypothess that both group correlatons are and the alternatve hypothess that the correlaton n group 2 s usng a two-sded z test (whch uses Fsher's z-transformaton) wth a sgnfcance level of Ths report shows the values of each of the parameters, one scenaro per row. The defntons of each column are gven n the Report Defntons secton of the report, so they wll not be repeated here. The values from ths table are plotted n the chart below. Plots Secton These plots show the relatonshp between alpha, power, and sample sze n ths example NCSS, LLC. All Rghts Reserved.

9 PASS Sample Sze Software Tests for Two Correlatons Example 2 Fndng the Sample Sze Contnung wth the prevous example, suppose the researchers want to determne the exact sample sze necessary to acheve 90% power at a 0.05 sgnfcance level. Setup Ths secton presents the values of each of the parameters needed to run ths example. Frst, from the PASS Home wndow, load the Tests for Two Correlatons procedure wndow by clckng on Correlaton, then Tests for Two Correlatons. You may then make the approprate entres as lsted below, or open Example 2 by gong to the Fle menu and choosng Open Example Template. Opton Value Desgn Tab Solve For... Sample Sze Alternatve Hypothess... Ha: ρ1 ρ2 Power Alpha Group Allocaton... Equal (N1 = N2) ρ1 (Correlaton Group 1) ρ2 (Correlaton Group 2) Output Clck the Calculate button to perform the calculatons and generate the followng output. Numerc Results Numerc Results when Ha: ρ1 ρ2 Target Actual Power Power N1 N2 N ρ1 ρ2 ρ1 - ρ2 Alpha PASS has calculated the sample sze as 369 per group NCSS, LLC. All Rghts Reserved.

10 PASS Sample Sze Software Tests for Two Correlatons Example 3 Valdaton usng Zar Zar (1984) page 314 presents an example of calculatng the power for a test of two correlatons. In hs example, when N1 = 95, N2 = 98, ρ1 = 0.84, ρ2 = 0.78, and alpha = 0.05, the power s 22% for a two-sded test. Setup Ths secton presents the values of each of the parameters needed to run ths example. Frst, from the PASS Home wndow, load the Tests for Two Correlatons procedure wndow by clckng on Correlaton, then Tests for Two Correlatons. You may then make the approprate entres as lsted below, or open Example 1 by gong to the Fle menu and choosng Open Example Template. Opton Value Desgn Tab Solve For... Power Alternatve Hypothess... Ha: ρ1 ρ2 Alpha Group Allocaton... Enter N1 and N2 ndvdually N N ρ1 (Correlaton Group 1) ρ2 (Correlaton Group 2) Output Clck the Calculate button to perform the calculatons and generate the followng output. Numerc Results Numerc Results when Ha: ρ1 ρ2 Power N1 N2 N ρ1 ρ2 ρ1 - ρ2 Alpha PASS also calculates the power to be 22% NCSS, LLC. All Rghts Reserved.