Tests for Two Ordered Categorical Variables
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1 Chapter 253 Tests for Two Ordered Categorcal Varables Introducton Ths module computes power and sample sze for tests of ordered categorcal data such as Lkert scale data. Assumng proportonal odds, such data can be analyzed drectly wth a z-test, or usng logstc regresson or the Mann-Whtney test. The power and sample sze formulae presented here are consstent wth any of these analyss methods. The results used here were presented n a paper by Whtehead (993). They are also mentoned n the book by Julous (200) and Machn et al (997). Ordered categorcal data often results from surveys such as a qualty of lfe (QoL) survey n whch responses are categores such as very good, good, moderate, poor. When there are only two categores, an analyss usng two proportons should be used. When there are more than two responses, and those responses can be ordered, the technques descrbed n ths chapter can be used. Techncal Detals Suppose a varable has K possble responses C,...,CK. Further suppose that these categores can be ordered so that C s more desrable than C j f < j. Hence C s the best outcome and CK s the worst. Ths procedure compares the results from two groups whch wll be called control (C) and expermental (E). The number of respondents fallng wthn the th category of the control group s labeled N. The total number n the control group s N and n the expermental group s N 2. The total sample sze of the study s N = N + N2. Let p E denote the probablty that an ndvdual n the expermental group gves response probablty of an outcome of C or better. Thus Q E = p je j= group. Defne the log-odds rato for a partcular category as. Defne p and QE Q E θ = log =,, K. Q Q C, and let Q E be the Q smlarly for the control 253-
2 Ths measures the advantage of the expermental group over the control group. A postve θ ndcates that the expermental treatment s better than the control treatment. Proportonal Odds The proportonal odds model assumes that all of these log-odds ratos are equal to a common value θ. That s, the proportonal odds assumpton s that θ = = θk = θ. Thus the whole pattern of response dfferences can be summarzed a sngle parameter. The formulae to follow use the fact that effcent score Y s asymptotcally normally dstrbuted when θ s small and n s large. The test statstc and power formulae are as follows. Y = N + Y µ Y Z = σ Y k = N ( L U ) L = N j, = 2,..., K j= U K = N j, =,..., K j= + L C = U KC = 0 µ Y = θv 2 σ =V Y NN2N V = 3 3 ( N + ) 2 K 3 N N2N pe + p 2 ( ) N + = 2 K = N + N N 2 3 The null hypothess H : θ 0 (the two treatments are equvalent) can be tested aganst the alternatve 0 = H a : θ 0 by computng Z and rejectng f Z s greater than z α / 2. That s, P ( Z > z θ = 0) α / 2 α = The power s the probablty of rejectng a false null hypothess, thus the power for a specfed value θr s Power = P ( Z > z θ = θ ) = Φ α / 2 ( z θ V ) α / 2 If a one-sded test s needed, replace α / 2 wth α. R R 253-2
3 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. The parameters that may be selected are θ, Sample Sze (N), Sample Sze (N2), or Power. Test Null Hypothess Indcate whether the hypothess s one-sded or two-sded. If two-sded s selected, alpha s automatcally dvded by 2, so you do not need to half alpha. 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 θ = 0 when, n fact, θ > 0. Values must be between zero and one. Hstorcally, the value of 0.80 (Beta = 0.20) was used for power. Now, 0.90 (Beta = 0.0) 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 a true null hypothess s rejected. In ths procedure, a type-i error occurs when you reject the null hypothess that θ = 0 when n fact t s. 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 0.0 to 0.0 by 0.0. Note that when you are analyzng a two-sded test, you should enter alpha, not alpha/
4 Sample Sze (When Solvng for Sample Sze) Group Allocaton Select the opton that descrbes the constrants on N or N2 or both. The optons are Equal (N = 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. Enter N, solve for N2 Select ths opton when you wsh to fx N at some value (or values), and then solve only for N2. Please note that for some values of N, there may not be a value of N2 that s large enough to obtan the desred power. Enter N2, solve for N Select ths opton when you wsh to fx N2 at some value (or values), and then solve only for N. Please note that for some values of N2, there may not be a value of N that s large enough to obtan the desred power. Enter R = N2/N, solve for N and N2 For ths choce, you set a value for the rato of N2 to N, and then PASS determnes the needed N and N2, wth ths rato, to obtan the desred power. An equvalent representaton of the rato, R, s N2 = R * N. Enter percentage n Group, solve for N and N2 For ths choce, you set a value for the percentage of the total sample sze that s n Group, and then PASS determnes the needed N and N2 wth ths percentage to obtan the desred power. N (Sample Sze, Group ) Ths opton s dsplayed f Group Allocaton = Enter N, solve for N2 N s the number of tems or ndvduals sampled from the Group populaton. N 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 N 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
5 R (Group Sample Sze Rato) Ths opton s dsplayed only f Group Allocaton = Enter R = N2/N, solve for N and N2. R s the rato of N2 to N. That s, R = N2 / N. Use ths value to fx the rato of N2 to N whle solvng for N and N2. Only sample sze combnatons wth ths rato are consdered. N2 s related to N by the formula: N2 = [R N], where the value [Y] s the next nteger Y. For example, settng R = 2.0 results n a Group 2 sample sze that s double the sample sze n Group (e.g., N = 0 and N2 = 20, or N = 50 and N2 = 00). R must be greater than 0. If R <, then N2 wll be less than N; f R >, then N2 wll be greater than N. You can enter a sngle or a seres of values. Percent n Group Ths opton s dsplayed only f Group Allocaton = Enter percentage n Group, solve for N and N2. Use ths value to fx the percentage of the total sample sze allocated to Group whle solvng for N and N2. Only sample sze combnatons wth ths Group percentage are consdered. Small varatons from the specfed percentage may occur due to the dscrete nature of sample szes. The Percent n Group must be greater than 0 and less than 00. 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 and to Group 2. The optons are Equal (N = 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 N and N2 ndvdually Ths choce permts you to enter dfferent values for N and N2. Enter N and R, where N2 = R * N Choose ths opton to specfy a value (or values) for N, and obtan N2 as a rato (multple) of N. Enter total sample sze and percentage n Group Choose ths opton to specfy a value (or values) for the total sample sze (N), obtan N as a percentage of N, and then N2 as N - N
6 Sample Sze Per Group Ths opton s dsplayed only f Group Allocaton = Equal (N = N2). The Sample Sze Per Group s the number of tems or ndvduals sampled from each of the Group and Group 2 populatons. Snce the sample szes are the same n each group, ths value s the value for N, 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. N (Sample Sze, Group ) Ths opton s dsplayed f Group Allocaton = Enter N and N2 ndvdually or Enter N and R, where N2 = R * N. N s the number of tems or ndvduals sampled from the Group populaton. N 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 N 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. R (Group Sample Sze Rato) Ths opton s dsplayed only f Group Allocaton = Enter N and R, where N2 = R * N. R s the rato of N2 to N. That s, R = N2/N Use ths value to obtan N2 as a multple (or proporton) of N. N2 s calculated from N usng the formula: where the value [Y] s the next nteger Y. N2=[R x N], For example, settng R = 2.0 results n a Group 2 sample sze that s double the sample sze n Group. R must be greater than 0. If R <, then N2 wll be less than N; f R >, then N2 wll be greater than N. 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. 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, mplctly defnes N 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 Ths opton s dsplayed only f Group Allocaton = Enter total sample sze and percentage n Group. Ths value fxes the percentage of the total sample sze allocated to Group. Small varatons from the specfed percentage may occur due to the dscrete nature of sample szes. The Percent n Group must be greater than 0 and less than 00. You can enter a sngle value or a seres of values
7 Effect Sze Control Group Category Proportons Ths opton contans the p, the proportons n each category of the control group. The number of categores s mpled by the number of tems n the lst. For example, an entry of mples that there are four categores and that 20% are category, 30% are category 2, 40% are category 3, and 0% are category 4. The frst category s the assumed to be the best outcome and the last category s the worst. Also, the categores are assumed to be ordered so that each s no better than the prevous category. For example, you mght have categores Very Good, Good, Neutral, Poor, Very Poor. Note that the values entered here should to one. If they do not, they are scaled so that they do. For example, the values (whch sum to 00) are scaled to (whch sum to one). You can enter any lst of postve, non-zero numbers. For example, f you wanted to ndcate that you have fve, equally-lkely categores, you could enter. θ (Log Odds Rato of Treatment Group) Enter one or more values of θ, the logarthm of the odds. Ths value represents the effect sze to be detected by the study. Note that the log odds rato s assumed to be constant for all categores. Examples to.4 by
8 Example Fndng the Power Suppose a clncal tral s planned to compare the response, made by a doctor, to certan treatment. The subjects are dvded nto two groups: those that wll receve the current treatment and those that wll receve an expermental treatment. Three months after the admnstraton of the treatment, the doctor rates the response as very good, good, moderate, or poor. Hstorcally, the responses have been about 20% very good, 50% good, 20% moderate, and 0% poor. The researchers want to consder a range of possble value of θ from 0.5 to 2.0. They want to look at the power acheved by sample szes from 30 to 50 per group. They want to set alpha to 0.05 and analyze the results wth 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 procedure wndow by expandng Proportons, then clckng on Ordered Categorcal, and then clckng on. You may then make the approprate entres as lsted below, or open Example by gong to the Fle menu and choosng Open Example Template. Opton Value Desgn Tab Solve For... Power Null Hypothess... Two-Sded Alpha Group Allocaton... Equal (N = N2) Sample Sze Per Group to 50 by 0 Control Group Category Proportons θ (Log Odds Rato of Treatment Group). 0.5 to 2.0 by 0.5 Annotated Output Clck the Calculate button to perform the calculatons and generate the followng output. Numerc Results Numerc Results for Two-Sample, Ordered-Categorcal Test Null Hypothess: θ = 0 Alternatve Hypothess: θ 0 Trt Control Trt Log Odds Categ. Categ. Rato Prop. Prop. Power N N2 N θ PC() PE() Alpha
9 References Whtehead, John 'Sample Sze Calculatons for Ordered Categorcal Data.' Statstcs n Medcne, 2, Julous, Steven A Sample Szes for Clncal Trals. Chapman & Hall/CRC. Boca Raton, Fl. Machn, D., Campbell, M., Fayers, P., and Pnol, A Sample Sze Tables for Clncal Studes, 2nd Edton. Blackwell Scence. Malden, Mass. Report Defntons Power s the probablty of rejectng a false null hypothess. N and N2 are the number of subjects n the frst and second groups. N s the total sample sze, N + N2. θ s log odds rato = log[odds(expermental)/odds(control)]. PC() Control Cat. Prop. s the proporton or probablty of the frst (best) category of the control group. PE() Trt Cat. Prop. s the proporton or probablty of the frst (best) category of the treatment group. Alpha s the probablty of rejectng a true null hypothess. Summary Statements Samples of 30 subjects n the control group and 30 subjects n expermental group acheve 7% power to detect a change n the log odds rato (θ) of when the sgnfcance level (alpha) s usng a two-sded test. Ths report shows the numerc results of ths power study. Followng are the defntons of the columns of the report. Power Ths s the probablty of rejectng a false null hypothess. N, N2 Ths s the number of subjects n the control group and expermental group, respectvely. Log Odds Rato Ths s the log of the odds rato. It s calculated usng the formula QE Q E θ = log =,, K. Q Q Alpha Ths s the probablty of rejectng a true null hypothess. Ths s often called the sgnfcance level. Beta Ths s the probablty of acceptng a false null hypothess. Control Cat. Prob. PC() Ths s the value of p C, the probablty of a control group subject respondng n category. Exp l Cat. Prob. PC() Ths s the value of p E, the probablty of an expermental group subject respondng n category
10 Category Probablty Dstrbuton Category Probablty Dstrbuton θ Pr() Pr(2) Pr(3) Pr(4) Ths report shows the ndvdual response probabltes. The frst row contans the results for the control group where θ s zero. The next row gves the values of p for each value of θ. The goal of the report s to let you E study the mpact on the pe of each value of θ. You can make ths assessment by watchng how much these values change over the correspondng value n the frst row. Plots Secton 253-0
11 Ths plot gves a vsual presentaton to the results n the Numerc Report. 253-
12 Example 2 Fndng the Sample Sze Contnung wth Example, the researchers want to fnd the sample sze necessary to acheve 90% power when θ s 0.9. Setup Ths secton presents the values of each of the parameters needed to run ths example. Frst, from the PASS Home wndow, load the procedure wndow by expandng Proportons, then clckng on Ordered Categorcal, and then clckng on. 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 Null Hypothess... Two-Sded Power Alpha Group Allocaton... Equal (N = N2) Control Group Category Proportons θ (Log Odds Rato of Treatment Group). 0.9 Output Clck the Calculate button to perform the calculatons and generate the followng output. Numerc Results Numerc Results for Two-Sample, Ordered-Categorcal Test Null Hypothess: θ = 0 Alternatve Hypothess: θ 0 Trt Control Trt Log Odds Categ. Categ. Rato Prop. Prop. Power N N2 N θ PC() PE() Alpha The requred sample sze s 92 n each group
13 Example 3 Valdaton usng Whtehead (993) Whtehead (993) has an example n whch he calculates the sample sze to be 94 when θ s 0.887, alpha s 0.05, power s 90%, the control group proportons are 0.2, 0.5, 0.2, and 0.. Setup Ths secton presents the values of each of the parameters needed to run ths example. Frst, from the PASS Home wndow, load the procedure wndow by expandng Proportons, then clckng on Ordered Categorcal, and then clckng on. You may then make the approprate entres as lsted below, or open Example 3 by gong to the Fle menu and choosng Open Example Template. Opton Value Desgn Tab Solve For... Sample Sze Null Hypothess... Two-Sded Power Alpha Group Allocaton... Equal (N = N2) Control Group Category Proportons θ (Log Odds Rato of Treatment Group) Output Clck the Calculate button to perform the calculatons and generate the followng output. Numerc Results Numerc Results for Two-Sample, Ordered-Categorcal Test Null Hypothess: θ = 0 Alternatve Hypothess: θ 0 Trt Control Trt Log Odds Categ. Categ. Rato Prop. Prop. Power N N2 N θ PC() PE() Alpha PASS matches the requred sample sze of 94 per group
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