AP Statistics Curriculum 2007 MultiVar ANOVA - Revision history

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2020-03-03T20:23:58Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

Dinov: /* MANOVA usage */

2014-10-20T13:02:46Z

‎MANOVA usage

IvoDinov: /* Hypotheses testing and inference */

2009-03-02T21:31:47Z

‎Hypotheses testing and inference

IvoDinov: /* Roy’s Largest (or Characteristic) Root */

2009-03-02T21:18:01Z

‎Roy’s Largest (or Characteristic) Root

IvoDinov at 21:16, 2 March 2009

2009-03-02T21:16:56Z

IvoDinov: /* Motivational example */

2009-03-02T20:39:27Z

‎Motivational example

IvoDinov: /* Motivational example */

2009-03-02T20:37:25Z

‎Motivational example

IvoDinov: /* ANOVA/MANOVA parallels */

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‎ANOVA/MANOVA parallels

IvoDinov: /* Variance partitioning */

2009-03-02T20:34:41Z

‎Variance partitioning

IvoDinov: /* Variance partitioning */

2009-03-02T20:32:33Z

‎Variance partitioning

← Older revision		Revision as of 20:23, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

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 ===MANOVA usage===
 There are two major situations in which MANOVA is appropriate.
-* When there are several correlated dependent variables, and the investigator needs a single, overall
+* When there are several correlated dependent variables, and the investigator needs a single, overall statistical test on this set of variables instead of performing multiple individual tests.
 * To explore how independent variables influence some patterning of response on the dependent variables.

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 There 4 major types of MANOVA tests. The ''statistical power'' of these tests differ and follow this rule:
 : '''Pillai’s > Wilk's > Hotelling’s > Roy’s Robustness'''.
 With the Pillai’s test being the most powerful. In general, however, all 4 omnibus tests will agree.
 ===Wilk’s <math>\Lambda</math>===
 This is the most popular test, which expressed as the ratio of determinant-error to determinant-total and represents percent variance not explained in the model. <math>1-\Lambda</math> is an index of variance explained
 by the model. <math>\eta^2</math> is a measure of effect size analogous to <math>R^2</math> in regression. You want the <math>\Lambda</math> value to be small, and this value is transformed into an approximate F for hypothesis-testing. Wilk’s <math>\Lambda</math> is the pooled ratio of error variances to effect variance plus error variance.
 ===Pillai’s Trace===
 Pillai’s criterion is the pooled effect variances.
 ===Hotelling’s Trace===
 Hotelling’s trace is the pooled ratio of effect variance to error variance.
 ===Roy’s Largest (or Characteristic) Root===
-The Roy's largest root gives an upper bound for the F statistic. Roy's largest root = <math>\max(\lambda_i)</math>, or the maximum eigenvalue of <math>A = HE^{-1}</math>. As a ''root'' is another name for an eigenvalue, this statistic is also called ''Roy's largest eigenvalue''.
+The Roy's largest root gives an upper bound for the F statistic.
 <hr>

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 ===Roy’s Largest (or Characteristic) Root===
-The Roy's largest root gives an upper bound for the F statistic. Roy's largest root = max(li), or the maximum eigenvalue of <math>A = HE^{-1}</math>. As a ''root'' is another name for an eigenvalue, this statistic is also called ''Roy's largest eigenvalue''.
+The Roy's largest root gives an upper bound for the F statistic. Roy's largest root = <math>\max(\lambda_i)</math>, or the maximum eigenvalue of <math>A = HE^{-1}</math>. As a ''root'' is another name for an eigenvalue, this statistic is also called ''Roy's largest eigenvalue''.
 <hr>

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 * To identify the independent variables which differentiate a set of dependent variables the most.
-Multiple analysis of covariance (MANCOVA) is similar to MANOVA, but allows adding of interval independents as ''covariates''. These covariates serve as control variables for the independent factors, serving to reduce the error term in the model. Like other control procedures, MANCOVA can be seen as analysis asking ''what would happen if all cases scored equally on the covariates to enable isolation of the effect of the factors beyond the covariates.''
+Multiple analysis of covariance (MANCOVA) is similar to MANOVA, but allows adding of interval independents as ''covariates''. These covariates serve as control variables for the independent factors, serving to reduce the error term in the model. Like other control procedures, [http://en.wikipedia.org/wiki/MANCOVA MANCOVA] can be seen as analysis asking ''what would happen if all cases scored equally on the covariates to enable isolation of the effect of the factors beyond the covariates.''
 ==[[EBook#Chapter_XI:_Analysis_of_Variance_.28ANOVA.29 | ANOVA]]/MANOVA parallels==
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   \end{pmatrix} </math>.
-Where the covariance between variables 1 and 2 is expressed in terms of their correlation (<math>\rho_{1,2}</math>) and individual variances (<math>\sigma_1</math> and <math>\sigma_2</math>). Under the null hypothesis, the scores for all subjects in groups 1, 2 and 3 are sampled from the same distribution. [http://ibgwww.colorado.edu/~carey/p7291dir/handouts/manova1.pdf You can see the complete arithmetic details here].
+Where the covariance between variables 1 and 2 is expressed in terms of their correlation (<math>\rho_{1,2}</math>) and individual variances (<math>\sigma_1</math> and <math>\sigma_2</math>). Under the null hypothesis, the scores for all subjects in groups 1, 2 and 3 are sampled from the same distribution. You can see the complete arithmetic details [http://ibgwww.colorado.edu/~carey/p7291dir/handouts/manova1.pdf here] and [http://userwww.sfsu.edu/~efc/classes/biol710/manova/manova.htm here].
 ==Computational Resources: Internet-based SOCR Tools==
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 Currently  under development.
-==Hands-on Activities==
+==Hypotheses testing and inference==
-Currently  under development.
+There 4 major types of MANOVA tests. The ''statistical power'' of these tests differ and follow this rule:
 <hr>

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 ===Motivational example===
-This examine will help us motivate and understand the MANOVA design. Suppose a brain mapping researcher is interested in exploring the treatment efficacy of dementia for randomly assigned patients into four conditions:
+This examine will help us motivate and understand the MANOVA design. Suppose a brain mapping researcher is interested in exploring the treatment efficacy of dementia for randomly assigned patients into four conditions ([http://www.stat.ucla.edu/%7Edinov/courses_students.dir/04/Spring/Stat233.dir/HWs.dir/AD_NeuroPsychImagingData1.html you can find such neuroimaging data here]):
 * A '''placebo control group''' who received ''physical therapy'' and a ''placebo drug'';
 * A '''placebo cognitive therapy group''' who received the placebo medication and a cognitive therapy;

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 * ''Main Effects'':
 ** Therapy: The univariate ANOVA main effect for therapy tells whether the physical vs. cognitive therapy groups have different means, irrespective of their medications. The MANOVA main effect for therapy tells whether the physical vs. cognitive therapy group have different mean vectors irrespective of their medication. The vectors in this case are the (3 x 1) column vectors of means (MMSE, CDR and Imaging).
-** Medication: The univariate ANOVA for medication tells whether the placebo group has a different mean
+** Medication: The univariate ANOVA for medication tells whether the placebo group has a different mean from the Vitamin-E group irrespective of the therapy type. The MANOVA main effect for medication tells whether the placebo group has a different mean vector from the VItamin-E group irrespective of therapy.
 * ''Interaction Effects'': The univariate ANOVA interaction tells whether the four means for a single variable differ from the value predicted from knowledge of the main effects of therapy and medication. The MANOVA interaction term tells whether the four mean vectors differ from the vector predicted from knowledge of the main effects of therapy and medication.

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 ==[[EBook#Chapter_XI:_Analysis_of_Variance_.28ANOVA.29 | ANOVA]]/MANOVA parallels==
-The purpose of the [[EBook#Chapter_VII:_Point_and_Interval_Estimates |T-test is to assess the likelihood that the means for ''two groups'' are sampled from the same sampling distribution of means. The purpose of an [[AP_Statistics_Curriculum_2007_ANOVA | ANOVA]] is to test whether the ''means for two or more groups'' are taken from the same sampling distribution. The multivariate equivalent of the T-test is [http://en.wikipedia.org/wiki/Hotelling%27s_T-square_distribution Hotelling’s T2]. Hotelling’s T2 tests whether the ''two vectors of means for the two groups'' are sampled from the same sampling distribution. MANOVA is the multivariate analogue to Hotelling's T2. '''The purpose of MANOVA is to test whether the vectors of means for the two or more groups are sampled from the same sampling distribution'''. Just as Hotelling's T2 will provide a measure of the likelihood that two random vectors of means come from the same distribution, MANOVA gives a measure of the overall likelihood of observing two or more random vectors of means out of the same distribution.
+The purpose of the [[EBook#Chapter_VII:_Point_and_Interval_Estimates |T-test is to assess the likelihood that the means for ''two groups'' are sampled from the same sampling distribution of means. The purpose of an [[EBook#Chapter_XI:_Analysis_of_Variance_.28ANOVA.29 | ANOVA]] is to test whether the ''means for two or more groups'' are taken from the same sampling distribution. The multivariate equivalent of the T-test is [http://en.wikipedia.org/wiki/Hotelling%27s_T-square_distribution Hotelling’s T2]. Hotelling’s T2 tests whether the ''two vectors of means for the two groups'' are sampled from the same sampling distribution. MANOVA is the multivariate analogue to Hotelling's T2. '''The purpose of MANOVA is to test whether the vectors of means for the two or more groups are sampled from the same sampling distribution'''. Just as Hotelling's T2 will provide a measure of the likelihood that two random vectors of means come from the same distribution, MANOVA gives a measure of the overall likelihood of observing two or more random vectors of means out of the same distribution.
 ===MANOVA usage===

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 the <math>V_{therapy}</math> matrix. The variance of CDR calculated from a univariate ANOVA is the second
 diagonal element in <math>V_{therapy}</math>. The variance in Imaging due to the interaction between
-therapy and medication, as calculated from a univariate ANOVA, will be the third diagonal element in <math>V_(therapy * medication)</math>. The off diagonal elements are all covariances and should be interpreted as ''between-group covariances''. So, in <math>V_{therapy}</math>, COV(1,2) = COV(MMSE, CDR) tells us whether the physical therapy group with the highest mean score on MMSE also has the highest mean score on the CDR. If, cognitive therapy were more efficacious than the physical therapy, then we should expect that all the covariances in <math>V_{therapy}</math> be large and positive.
+therapy and medication, as calculated from a univariate ANOVA, will be the third diagonal element in <math>V_{therapy * medication}</math>. The off diagonal elements are all covariances and should be interpreted as ''between-group covariances''. So, in <math>V_{therapy}</math>, COV(1,2) = COV(MMSE, CDR) tells us whether the physical therapy group with the highest mean score on MMSE also has the highest mean score on the CDR. If, cognitive therapy were more efficacious than the physical therapy, then we should expect that all the covariances in <math>V_{therapy}</math> be large and positive.
-Similarly, COV(2,3) = cov(CDR, Imaging) in the <math>V_(therapy * medication)</math> matrix has the following interpretation: if we control for the main effects of therapy and medication, then do groups with high average CDR scores also tend to have high average scores on the brain atrophy as reported by Imaging observations? If there were no main effect for therapy on any of the measures, then all the elements of <math>V_{therapy}</math> will be 0. If there were no main effect for medication, then <math>V_{medication}</math> will be all 0's. Finally, if there were no interaction, then all of <math>V_(therapy * medication)</math> would be 0's. It makes sense to have these matrices manually inspected as they may provide such valuable information about trivial effects or interactions.
+Similarly, COV(2,3) = cov(CDR, Imaging) in the <math>V_{therapy * medication}</math> matrix has the following interpretation: if we control for the main effects of therapy and medication, then do groups with high average CDR scores also tend to have high average scores on the brain atrophy as reported by Imaging observations? If there were no main effect for therapy on any of the measures, then all the elements of <math>V_{therapy}</math> will be 0. If there were no main effect for medication, then <math>V_{medication}</math> will be all 0's. Finally, if there were no interaction, then all of <math>V_{therapy * medication}</math> would be 0's. It makes sense to have these matrices manually inspected as they may provide such valuable information about trivial effects or interactions.
 ==Approach==

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 The diagonal elements (<math>V_{error_j}</math>) are the error variances, for each type of response, and have the same meanings as the error variances in their univariate ANOVA counterparts (an average variability within the four groups). The error variance in MANOVA has the same meaning. That is, if we did a univariate ANOVA for the MMSE alone, the error variance would be the mean squares within groups. <math>V_{error1}</math> would be the same means squares within groups -- it would be the same number as in the univariate analysis. The same holds for the mean squares within groups for the CDR and Imaging. The only difference is in the off diagonal elements, which represent the covariances for
 the error matrix and must be interpreted as within-group covariances. That is, ''COV(error1,error2)'' tells us the extent to which individuals within a group who have high MMSE scores also tend
-to have high CDR scores. This covariance matrix can be scaled to ''correlations'' to ease the inspection. For example, <math>Corr(error1,error2) = {COV(error1,error2) \over {V_{error1}\times V_{error2}}^{1\over2}}</math>.
+to have high CDR scores. This covariance matrix can be scaled to ''correlations'' to ease the inspection. For example, <math>Corr(error1,error2) = {COV(error1,error2) \over {(V_{error1}\times V_{error2})}^{1\over2}}</math>.
 When analyzing real data, you always have this error covariance matrix and its correlation