Lecture 6: Generalized multivariate analysis of variance

Transcription

1 Lecture 6: Generalize multivariate analysis of variance Measuring association of the entire microbiome with other variables Distance matrices capture some aspects of the ata (e.g. microbiome composition, relative abunance, phylogenetic relationships). Eucliean istance (square-root of sums of square ifferences between components of the centere ata) captures the covariances of the variables. Can these characteristics be use to raw association of the entire microbiome with other variables of interest (e.g. treatment group, locus of sampling, etc.)? cecal fecal

2 Measuring association of the entire microbiome with other variables Distance matrices capture some aspects of the ata (e.g. microbiome composition, relative abunance, phylogenetic relationships). Eucliean istance (square-root of sums of square ifferences between components of the centere ata) captures the covariances of the variables. Can these characteristics be use to raw association of the entire microbiome with other variables of interest (e.g. treatment group, locus of sampling, etc.)? 3 A general strategy for multivariate analysis Apply a normalization to the ata (e.g. relative abunance); Calculate a istance metric between the observations (e.g. Unifrac, Jensen-Shannon, Chi-Square); Perform orination an/or clustering analysis to visualize relationships between observations; Test for ifferences between preefine groups (e.g. treatment levels, phenotypes) 4

3 ANOVA Iea: SS total = SS error + SS treatments F test: F = [SS treatments /(I )]/[SS error /(n T - I)] F = (variance between)/(variance within treatments) I number of treatments n T total number of cases ANOVA example a a a SS SS SS 3 (6 Y ) =(6 )^ = 9 (8 ) = 9 9 (4 ) = 0 ( ) = (3 ) = (4 ) = 4. Within group means Y = ( )/6 = Y = = 9 Y 3 = =. Overall mean Y = 8 3. Between group sum of squares SS treatments = n (Y -Y) +n (Y -Y) +n 3 (Y 3 -Y) = 84 (k ) = 3 = 4. Within group sum of squares SS error = 68 (nt k) = 8 3 =. F = (84/) / (68/) = 4/4. = Fcritical (, ) = Conclusion: The group effects are statistically significantly ifferent. 8. Next: perform post-hoc pairwise tests to etect the pairs that are ifferent 6 3

4 Eucliean MANOVA A irect extension of the univariate ANOVA to multiple variables. SS = Σ(Y i Y) T (Y i Y) SS = Σ, where is the Eucliean istance from the center. 7 Geometric representation of MANOVA (Anerson, 00) SS A /(a ) F SSW /(N a) SS A between group sums of squares SS W within group sums of squares SS T total sum of squares Fig.. A geometric representation of MANOVA for two groups in two imensions where the groups iffer in location. The within-group sum of squares is the sum of square istances from iniviual replicates to their group centroi. The among-group sum of squares is the sum of square istances from group centrois to the overall centroi. ( ) Distances from points to group centrois; ( ) istances from group centrois to overall centroi; ( ), overall centroi; ( ), group centroi; ( ), iniviual observation. SS T = SS W + SS A Key: Mean within group square istance is equal to sum of square istances to the centroi. 8 4

5 Calculating F-statistic from arbitrary istance matrices SS A /(a ) F SSW /(N a) Fig. 3. Schematic iagram for the calculation of (a) a istance matrix from a raw ata matrix an (b) a non-parametric MANOVA statistic for a one-way esign (two groups) irectly from the istance matrix. SST, sum of square istances in the half matrix ( ) ivie by N (total number of observations); SSW, sum of square istances within groups ( ) ivie by n (number of observations per group). SSA SST SSW an F = [SSA/(a )]/[SSW/ (N a)], where a the number of groups. 9 Obtaining p-values The F-statistic oes not follow Fisher s F-ratio uner null, therefore we nee to evaluate it s istribution uner null. Null hypothesis: there is no ifference between groups; therefore, we can compute null istribution empirically by shuffling the group labels. For each reshuffling of labels compute F statistic, the p-value is then (No. of F F) P (Total no. of F )

6 Post-hoc tests for multi-level factors When a factor has more than levels, it is not immeiately clear which pair of groups are ifferent from each other. To figure this out a post-hoc pairwise tests nee to be carrie out. Pairwise p-values are calculate with aitional permutations. Multiple comparison correction may be necessary. More sophisticate esigns Two-way MANOVA Straightforwar extension with all interactions consiere. Stratification/block esign When an effect is to be etermine within the levels of another factor E.g. Location of sampling vs. treatment 6

7 More sophisticate regression scenarios Base on Zapala & Schork, PNAS 006. Suppose we have M preictor variables We treat the multivariate (N P) ata (microbiome abunance, gene expression, etc.) as the response variable Y The basic multivariate regression moel is Y = Xβ + ε, where β is the coefficient matrix, an ε is an error term. Define the hat matrix as usual H = X + X, X. 3 Regression scenario (continue) G = I n + D () I n + ; Then F = ;<(HGH)/(>,) ;< I,H G I,H /(?,>). This is how PERMANOVA is implemente in R/vegan package, function aonis(). 4 7

8 Assumptions of PERMANOVA PERMANOVA is efine for balance sample sizes, but can be rewritten for n A n C. Homosceasticity is an unerlying assumption. Do violations of these assumptions lea to unesire behaviors? Simulation to test these asumptions: Let X be,000 imensional uncorrelate stanar normal Let Y be,000 imensional uncorrelate multivariate normal with each component mean = /sqrt(00)*e S.D. = 0.8 Simulate ata with n A, n C,,,0 Compute Eucliean istances, PERMANOVA p-values Empirical robustness of PERMANOVA to heterosceasticity an unbalance sample sizes Fraction of rejecte null hypotheses Effect: 0 Effect: Effect: 4 Effect: Empirical type I error an power (α = 0.0) Observations in most isperse sample Balance sample size No Yes Observations in least isperse sample 0 Type I error 6 8

9 Robustness of PERMANOVA When both homosceasticity an balance sample sizes are violate averse statistical behavior can be observe. If X is the more isperse sample then n A < n C leas to type I error inflation, n A > n C leas to loss of power, where n A is the number of observations in the more isperse sample. 7 Iea: Univariate approach to heterosceasticity issues Consier the square of Welch t-statistic T M = A,CO P Q P R S R UQ P V WS V If we can write T M in terms of pairwise istances, we can generalize it to multivariate ata. We can use permutation testing to assess the significance.. 8 9

10 Key equations for T M erivation s A = S R x S R S R, [ x \ = S R S R S R, [\, Where S enotes ouble summation S S Let Z = (z,, z SR US V ) = (x,, x SR, y,, y SV ), x yo = S RUS V S R S V [` \ `[U. e z S R US [ z \ V x S [ x \ R y S [ y \ V f. 9 Pseuo-F vs T M S R US V n A + n [\ C S R n [\ A S R US V n [\ C [,\` [,\ ` [,\`S F = R U S R n [\ + S R US V A n [\ /(n A n C ) C T M = n A + n C n A n C [,\` [,\`S R U S R US V n A + n [\ C S R n [\ S R US V A n [\ C [,\` [,\` [,\`S R U n A (n A ) S R [\ + n C (n C ) S R US V [\ [,\` [,\`S R U How o these compare when n A = n C or S R (S R, ) [\? S V (S V,) [\ 0

11 7/6/6 Effect: 0 Effect: 4 Effect: Fraction of rejecte null hypotheses 0. Type I error Empirical type I error an power (α = 0.0) Frac.S.D.: 0. Frac.S.D.: 0.8 Frac.S.D.: Balance sample size No Yes Statistical test PERMANOVA Tw Observations in least isperse sample Empirical performance of T M vs PERMANOVA Homosceastic 0 0 Observations in most isperse sample 0 Typical experimental scenarios at n or n 0 Fraction of rejecte null hypotheses Effect: 0 Effect: 4 Effect: 0.0 Observations 0. in least isperse sample Balance sample size 0. No Yes Statistical test PERMANOVA Tw Empirical type I error an power (α = 0.0) Observations in most isperse sample Frac.S.D.: 0. Frac.S.D.: 0.8 Frac.S.D.: Fraction of rejecte null hypotheses Effect: 0 Effect: Effect: Empirical type I error an power (α = 0.0) Observations in most isperse sample 0 Frac.S.D.: 0. Frac.S.D.: 0.8 Frac.S.D.: 0.99 Balance sample size No Yes Statistical test PERMANOVA Tw Observations in least isperse sample

12 Performance in a real ataset Table. Comparison of PERMANOVA an T W on mouse gut microbiome ataset. Cecal microbiome P-values Fecal microbiome P-values Comparison N obs. H! PERMANOVA T W N obs. H! PERMANOVA T W C. vs. All Abx. vs vs C. vs. Penicillin vs vs C. vs. Vancomycin vs vs C. vs. Tetracycline vs vs C. vs. Van. + Tetr. vs vs PERMANOVA-S: accommoates multiple istances Base on Tang et al. Bioinformatics 06. Suppose we want to consier K istances simultaneously, D,, D i. We woul like to know the significance of the entire ensemble Determine which iniviual istance performs best 4

13 PERMANOVA-S: Ensembling algorithm. For each D j, compute the observe pseuo-f statistic F j ;. Obtain B permutations an compute F j (),, F j k ; 3. Compute p-value for each k, p j, an p m[s = min (p,, p i ); 4. For each k, compute the permutation p-value as p (q) j = (B rank(f q v )/B;. For each permutation b, obtain minimal permutation p-value (q) p m[s = min (p q,, p q j ). 6. The final (unifie) p-value is the proportion of p () q m[s,, p m[s smaller than p m[s. Summary PERMANOVA is useful for omnibus hypothesis testing; PERMANOVA has unesirable behavior with unbalance heterosceastic ata; T M corrects that behavior in two sample case; PERMANOVA testing can be one with ensembling multiple istances. 6 3