Postgraduate course: Anova Repeated measurements Day (part ) Postgraduate course in ANOVA Repeated Measurements Day (part ) Summarizing homework exercises. Nielrolle Andersen Dept. of Biostatistics, Aarhus University Exercise : C- reactive protein Using the MANOVA we found a significant difference between the groups with respect to changes over time p=.9). We found a statistical significant changes over time in the -group (p=.7) in the -group (p=.). Assumptions fulfilled? Exercise : C- reactive protein (ln-transformed).. Helmut. Using the MANOVA we found no significant difference between the groups with respect to changes over time p=.7). We found a statistical significant changes over time in the -group (p=.) in the -group (p=.7). We found a significant difference between levels (the distance between the parel curves) in the groups (p=.). Assumptions fulfilled? Univariate Repeated Measurements ANOVA:original data anova creac_p group/id group time time*group, repeated(time) Parts of the output: Lowest b.s.e. variable: id Between-subjects error term: id group Levels: ( df) Covariance pooled over: group (for repeated variable) Repeated variable: time Huynh-Feldt epsilon =. Greenhouse-Geisser epsilon =.7 Box's conservative epsilon =. time..... time*group 7....7. Residual We found a significant difference between the groups with respect to changes over time p<. or =.).
Postgraduate course: Anova Repeated measurements Day (part ) Univariate Repeated Measurements ANOVA (ln-tansformed data): anova lcreac_p group/id group time time*group, repeated(time) Parts of the output: Between-subjects error term: id group Lowest b.s.e. variable: id Levels: ( df) Covariance pooled over: group (for repeated variable) Repeated variable: time Huynh-Feldt epsilon =.7 *Huynh-Feldt epsilon reset to. Greenhouse-Geisser epsilon =.79 Box's conservative epsilon =. time..... time*group..... Residual Vo significant difference between the groups with respect to changes over time p>. or. anova lcreac_p group/id group time, repeated(time) Parts of output Source Partial SS df MS F Prob > F Model.77.9.. group 7.7 7.7..9 id group..777 time.9979.97.9. Residual 7.7. Total.779 7.9 time.9.... Residual We found a statistical significant changes over time (p<.) We found a significant difference between levels (the distance between the apel curves) in the groups (p=.) Just as before. See next slide: The t test for comparing the sum between the groups:. generate float lcreac_p_sum= lcreac_p+lcreac_p+lcreac_p+lcreac_p. ttest lccreac_p_sum, by(group) 7 Estimates of the stard deviations correlations: Original data MS id con..7...7..979 Correlation. Group Obs Mean Std. Err. Std. Dev. [9% Conf. Interval].7.9. 9..9.7..7...9.7.79.7.79.77.9.7.7.9..9..7.7. combined.7.97.97.79.9 Ln transformed data MS id con Correlation diff -..79 -.79 -.97..9...79.97.. Degrees of freedom: Ho: mean() - mean() = diff = Ha: diff < Ha: diff!= Ha: diff >..7..7.79.7.7..77..9.7..9.7.7 t = -.97 t = -.97 t = -.97 P < t =. P > t =.9 P > t =.99 The differences between the groups becomes smer using the transformed data
Postgraduate course: Anova Repeated measurements Day (part ) 9 The probability plots for group : d_ - - - - dif d_ - - - - dif The probability plots for group : d_ - - - - dif - - - - Inv ers e N o r m al d_ - - - - dif - - - Inv er se N o r m al - - Inv erse N o rm al - - - dif d_ - - d_ - - dif d_ - - - - - - Inv ers e N o r m al d_ - - - - - - - - - Inv er se N o r m al ld_ ld_ -. - -. - -.. -. - - - - Inv erse N o rm al dif. -. -.. - -. -. -. - -. -. ld_ ld_ - -. - -. - -.. - - - dif - -.. dif -. -. - ld_ - -.. ld_ - -. - -.. dif -. -. -. -. Inv erse N o r m al -. - -.. Inv erse N o r m al ld_ - -.. ld_ - -. - -. dif - -.. Inv er se N o rm al dif - -. - -. Inv er se N o rm al The changes for each group : d_ - - - -.... G r ou p d_ - - - -.... G r o up Sum/difference plots, here -: No association between sum difference (changes) or between variation of changes sum in the ln-transformed data but maybe in original Yes, large change in group in the original data d_ - - ld_ - -.. ld_ - -. - -...... G r ou p.... G rou p.... G rou p ld_ ld_ d_ - - -. - -. - -.. -.... G r o up.... G ro up.... G ro up d_ - - c r ea c _p _ s u m ld_ - -. - -. c r ea c _ p_ s u m Conclusion: Use the ln-transforme data;the assumptions for MANOVA (almost) fulfilled for
Postgraduate course: Anova Repeated measurements Day (part ) Univariate Repeated Measurements ANOVA? bysort group: correlate lcreac_p lcreac_p lcreac_p lcreac_p, means -> group = (obs=) -- lcreac_p.97.777.9.9 lcreac_p.9.7..7979 lcreac_p.99.99.77.77 lcreac_p.77..7.97 lcreac~ lcreac~ lcreac~ lcreac~ -------------+------------------------------------ lcreac_p. lcreac_p.. lcreac_p..7. lcreac_p..7.7. group = (obs=) -- lcreac_p...97.9 lcreac_p..7..9 lcreac_p.7.79.9.79 lcreac_p..97.99.97 lcreac~ lcreac~ lcreac~ lcreac~ -------------+------------------------------------ lcreac_p. lcreac_p.7. lcreac_p... lcreac_p.... bysort group: correlate ld_ ld_ ld_, means -> group = (obs=) -- ld_ -.7.79 -.979. ld_ -.9.7 -.797. ld_ -.79. -.9 -.9 ld_ ld_ ld_ -------------+--------------------------- ld_. ld_.7. ld_.7.7. -> group = (obs=) -- ld_ -.997.77 -.9 ld_ -.77. -.9. ld_ -.77.97 -.9 -.7 ld_ ld_ ld_ -------------+--------------------------- ld_. ld_.9. ld_.7.7. It seems that Univariate Repeated Measurements ANOVA can be used. The C- reactive protein example. Result: We may describe the data with a figure showing the geomtric means (+/- sd from the ln-transformed data) with a table stating the three stard deviations (or coefficient of variation) the correlation for the two groups separately. SD CV.79.77...9..97.9.7.9..9...9... Correla -tion..7.7 correlat ion..7.7
Postgraduate course: Anova Repeated measurements Day (part ) At baseline: In the -group the geometric mean was.7 (9%CI.,., CV=.) in the sham group. (9%CI.,.7 CV=.). The ratio between the geometric means was.7 (9%CI.,. ) which was statisticy significant different from (p=.9). ttest lcreac_p, by(group) Group Obs Mean Std. Err. Std. Dev. [9% Conf. Interval].97.779.777.7.9..7..97. diff -.. -.99 -.99 Degrees of freedom: t = -.997 P > t =. Using the MANOVA we found no significant difference between the groups with respect to changes over time p=.). We found a common statistical significant changes over time(p<.), we estimated the ratio between the levels.7 (9%CI.,. ) which was statisticy significant different from (p=.)... ttest lccreac_p_sum, by(group) Divided by! diff -..79 -.79 -.97 Degrees of freedom: t = -.97 P > t =.9 Change from from baseline to end of study: In the -group the geometric mean increased by a factor. (9%CI.,.9) in the sham group the increase was. (9%CI.9,.). The ratio between the increases was. (9%CI.,.9) which was not statisticy significant different from (p=.). ttest ld_, by(group) Group Obs Mean Std. Err. Std. Dev. [9% Conf. Interval] -.79.7. -. -.9 -.77.79.97 -.7 -.7 diff.9.799 -..797 Degrees of freedom: t =.7 P > t =.9 If we use the Univariate Repeated Measurements ANOVA to estimate ratio s between the levels the common increase: The ratio between the levels is estimated to.7 (9%CI.7,. ) which was statisticy significant different from (p=.). (before:.7 (9%CI.,. ) p=.. ) The geometric means in both groups are estimated to increase by a factor. (9%CI.9,.9 ) which is statistical significant different from (p<.) (:. (9%CI.,.9), :. (9%CI.9,.).