Postgraduate course: Anova and Repeated measurements Day 2 (part 2) Mogens Erlandsen, Department of Biostatistics, Aarhus University, November 2010
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1 30 CVP (mean and sd) Postgraduate course in ANOVA and Repeated Measurements Day Repeated measurements (part ) Mogens Erlandsen Deptartment of Biostatistics Aarhus University The within subject variation is the relevant variation when analyzing changes over time..so How can we estimate the within subject variation and the between subject variation? 1 Univariate Repeated Measurements ANOVA using the anova command In order to use ANOVA we need stronger assumptions 3) The standard deviation σ T is the same for all measurements and the correlations between any two (different measurements) on the same subject are equal. i.e. σ T = σ B + σ W The correlation = σ B / σ T Note: The default behaviour in Stata s anova command is to test effects against the within subject standard deviation. This might be wrong if the effect is a between subjects effect. In this case Stata should be told. See next slide. 3 Example EVF continued (data in long format) Test 1: Hypothesis H: Parallel curves This test can be performed by a 3-way ANOVA with id (subject identification), time, and #time (interaction) in the model. is a between subjects effect Stata 11: anova evf /id time time#, repeated(time) The command wsanova (should be downloaded) might be easier: wsanova evf time, id(id) between() epsilon (allmost the same output!) set matsize 800, permanently before using the anova commands 4
2 The Univariate Repeated Measurements Anova: anova evf /id time time#, repeated(time) Output (continued) Between-subjects error term: id Levels: 30 (8 df) Lowest b.s.e. variable: id Covariance pooled over: (for repeated variable) Between subjects Within subjects Number of obs = 180 R-squared = Root MSE = Adj R-squared = Source Partial SS df MS F Prob > F Model id time time# Residual Total Test 1 5 Repeated variable: time Same as previous slide Huynh-Feldt epsilon = Greenhouse-Geisser epsilon = Box's conservative epsilon = Prob > F Source df F Regular H-F G-G Box time time# Residual 140 Some corrections of the p-value have been proposed when the assumptions (mainly assumption 3) are violated. They will normally be larger than the regular. 6 How can we use the four/three p - values: Prob > F Source df F Regular H-F G-G Box time time# Residual 140 The following has been proposed: If the regular/uncorrected p value is not significant (>0.05) then stop and accept (fail to reject) the hypothesis else If the G-G p value is significant (<0.05) then stop and reject the hypothesis else If the Box p value is significant (<0.05) then stop and reject the hypothesis else stop and accept (fail to reject) the hypothesis. wsanova evf time, id(id) between() epsilon Number of obs = 180 R-squared = Root MSE = Adj R-squared = Source Partial SS df MS F Prob > F Between subjects: id* Within subjects: time time* Residual Total Note: Within subjects F-test(s) above assume sphericity of residuals; p-values corrected for lack of sphericity appear below. Greenhouse-Geisser (G-G) epsilon: Huynh-Feldt (H-F) epsilon: Sphericity G-G H-F Source df F Prob > F Prob > F Prob > F time time* Kirk (198) 7 8
3 Test 3: (for each ): H4: no changes over time. wsanova evf time if ==1, id(id) epsilon Number of obs = 90 R-squared = Root MSE = Adj R-squared = Source Partial SS df MS F Prob > F id time Residual Total Note: Within subjects F-test(s) above assume sphericity of residuals; p-values corrected for lack of sphericity appear below. Greenhouse-Geisser (G-G) epsilon: Huynh-Feldt (H-F) epsilon: Sphericity G-G H-F Source df F Prob > F Prob > F Prob > F time wsanova evf time if ==, id(id) epsilon Number of obs = 90 R-squared = Root MSE = Adj R-squared = Source Partial SS df MS F Prob > F id time Residual Total Note: Within subjects F-test(s) above assume sphericity of residuals; p-values corrected for lack of sphericity appear below. Greenhouse-Geisser (G-G) epsilon: Huynh-Feldt (H-F) epsilon: Sphericity G-G H-F Source df F Prob > F Prob > F Prob > F time Lower bound for the p - value 9 10 If we want to estimate the within and between subject standard deviations (σ T = σ B + σ W ) one can use the xtmixed command: We have four variables: evf id time: xi: xtmixed evf i.time*i. id: ///,nofetable noheader no nostderr nolrtest Part of the output: Random-effects Parameters Estimate id: Identity sd(_cons) between subject sd sd(residual) within subject sd From xtmixed we have sd W = sd B = and the we can calculate s w = s B = sd T = sd B + sd W = s T = The (estimated) correlation between two measurements on the same subject s B / s T =
4 We can look at each separately: xi: xtmixed evf i.time id: if ==1 ///,nofetable noheader no nostderr nolrtest Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] id: Identity sd(_cons) sd(residual) xi: xtmixed evf i.time id: if == ///,nofetable noheader no nostderr nolrtest Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] id: Identity sd(_cons) sd(residual) Estimates in each : grp 1 s W s B s T s W s B s T We can see that the estimates for the between subject variation (s B ) are almost equal but the within subject variation (s W ) are different and hence also the total variation and the correlation. Remarks: The correlations are expected to be positive (why?), but in special cases one might get negative correlations. (weight of mice with limit amount of food and.) We can compare the estimates above with the standard deviations and correlation calculated from the 6 variables evf1, evf,.., evf Conclusion: We found a significant differerence between the s with respect to changes over time p<0.004) We found a statistical significant changes over time in the CPB- (p<0.006) but not in the Sham- (p>0.19) grp 1 s W s B s T Correlation Correlation Checking the model: A important part, but often suppressed, of the analysis is to check whether the assumptions for the analysis is fulfilled sufficiently (a weak statement ), or a transformation (ln-transformation??) of the data is better, or we need to look for an analysis with maybe weaker assumptions
5 Checking the model: The tests are normally F-tests and The result of the F-test is not affected by moderate departures from normality, especially for large numbers of observations in each. The F-test is more sensitive to the assumption of equal variances/ standard deviations.. unless the sample size in each are almost equal. (One can reduce the degrees of freedom as in the t test with unequal variance) Assumptions: Test 1: Parallel curves 1) All the differences between two timepoints are multivariate normaldistributed with in s. ) The sd s and the correlations between differences should be the same in the two s (mvtest) Test 3: No change over time (with-in a ) 1) All the differences between two timepoints are multivariate normaldistributed Example (evf): Probability plots for 1: Probability plots for : d1_ dif 1 d_3.15 dif dif 3 4 d1_ dif 1 d_3 dif dif Inver se Normal Inver se Normal dif 4 5 dif dif dif dif Inver se Normal dif Inver se Normal 19 0
6 Scatter plots for (some of) the differences: d_3 d1_ d1_ d_ The variation within the s should be equal for each set of differences (and all equal if we use the ANOVA) We can also use the figures from the paired analysis (see Basic Biostatistics ): difference (or changes) versus average (or sum). Bland-Altman plot Look for increasing (decreasing) changes when the average increase and/or increasing variation when the average increase If so then the ln-transformation of the data maybe appropriate. dif-ave plots: d_3 mvtest can also test for normality: d1_ d1_ d_ mvtest norm d1_ d_3 if ==1, stats(all) Test for multivariate normality Mardia mskewness = chi(35) = Prob>chi = Mardia mkurtosis = chi(1) = 0.61 Prob>chi = Henze-Zirkler = chi(1) = Prob>chi = Doornik-Hansen chi(10) = Prob>chi =
7 mvtest can also test for normality (bivariate): mvtest can also test for normality (univariate) mvtest norm d1_ d_3 if ==1, biv Doornik-Hansen test for bivariate normality Pair of variables chi df Prob>chi d1_ d_ d_ mvtest norm d1_ d_3 if ==1, uni Test for univariate normality joint Variable Pr(Skewness) Pr(Kurtosis) adj chi() Prob>chi d1_ d_ Conclusion: The assumptions (normality) seem to be ok; Similar result for Remark: be careful; a lot of tests 6 Assumption (The univariate (ANOVA) approach): 3) The standard deviation σ T is the same for all measurements and the correlations between any two (different measurements) on the same subject are equal.. mvtest cov evf1 evf evf3 evf4 evf5 evf6 if ==1, compound Test that covariance matrix is compound symmetric Adjusted LR chi(19) = 7.88 Prob > chi = mvtest cov evf1 evf evf3 evf4 evf5 evf6 if ==, compound Test that covariance matrix is compound symmetric Adjusted LR chi(19) = 7.7 Prob > chi = Conclusion: We accept the hypothesis for each 7 Checking the assumptions for the ANOVA approach : Group 1 Residuals Group Residuals Linear prediction Linear prediction Residuals Residuals residual probability-plot residual probability-plot
8 Conclusion: The evf-measurements seem to fulfilled the assumptions about the normal distribution but have problem with standard deviations/correlations between the s. The Univariate Repeated Measurement may be appropriate (for each sperately) and one can state the two/three standard deviation for the two s (i.e. in a figure showing the mean curves) (An analysis of ln-transformed data gives almost the same result) Example (distance): Distance time Boys Girls 9 30 Example (heartperiod): anova dist sex/idsex time sex#time, repeated(time) Number of obs = 108 R-squared = Root MSE = Adj R-squared = Example (distance): Between-subjects error term: idsex Levels: 7 (5 df) Lowest b.s.e. variable: id Covariance pooled over: sex (for repeated variable) Source Partial SS df MS F Prob > F Model sex idsex time sex*time Residual Total Repeated variable: time Huynh-Feldt epsilon = *Huynh-Feldt epsilon reset to Greenhouse-Geisser epsilon = Box's conservative epsilon = Prob > F Source df F Regular H-F G-G Box time sex*time Residual 75 Conclusion: We found no significant differerence between the s (sex) with respect to changes over time p>.078) 3
9 anova evf /id time time*, repeated(time) Number of obs = 108 R-squared = Root MSE = Adj R-squared = Source Partial SS df MS F Prob > F Model sex idsex time sex*time Residual Total If we accept H (parallel curves) we can test whether the two mean curves are equal. It is exactly the same test as day, part 1 i.e. equal to a t-test on the average of the 4 measurements of distances. All three assumptions should be fulfilled. 33 Example (distance): Between-subjects error term: idsex Levels: 7 (5 df) Lowest b.s.e. variable: id Covariance pooled over: sex (for repeated variable) Repeated variable: time Huynh-Feldt epsilon = *Huynh-Feldt epsilon reset to Greenhouse-Geisser epsilon = Box's conservative epsilon = Prob > F Source df F Regular H-F G-G Box time sex*time Residual 75 If we accept H (parallel curves) we can test H4 (no changes over time) for both s in one test. If we perform a test for each of the s we can have to different answers or we can accept H4 for both s separately due to low power. 34 If problems with the assumptions we can use a permutation test: permute sex r(f), reps(10000) :mvtest mean d8_10 d10_1 d1_14, by(sex) het.. Monte Carlo permutation results Number of obs = 7 command: mvtest mean d8_10 d10_1 d1_14, by(sex) het _pm_1: r(f) permute var: sex T T(obs) c n p=c/n SE(p) [95% Conf. Interval] _pm_ Note: confidence interval is with respect to p=c/n. Note: c = #{T >= T(obs)} Remarks: We have now more than one way to analyze the data. Which one (if any) shall we choose? How can describe the analysis? How can we describe the results? Depending of what we can assume we can try to answer the questions (Day 4). Conclusion: We reject H (p=0.030), the changes over time for the two s are statistical significant
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