• Alpha: Observed OTUs
    • Unconditional Model
    • Fixed Effect of Time
    • Fixed and Random effect of Time
    • Adding demographics
    • Intervention effect
  • Alpha: Shannon Diversity
    • Unconditional Model
    • Fixed Effect of Time
    • Fixed and Random effect of Time
    • Adding demographics
    • Intervention effect

Last updated: 2020-02-27

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This page contains the investigation of the changes in alpha diversity metrics (Observed and Shannon) over time.

Here, we investigated how the metrics of alpha diversity changed over time.

Alpha: Observed OTUs

Unconditional Model

ICC <- function(x){
  icc <- VarCorr(x)[[1]]/(VarCorr(x)[[1]] + sigma(x)**2)
  icc <- lapply(icc, function(x) { attributes(x) <- NULL; x })
  icc <- icc[[1]]
  return(icc)
}

mydata <- microbiome_data$meta.dat %>%
  mutate(intB = ifelse(Intervention=="B", 1,0),
         time = as.numeric(Week) - 1,
         female = ifelse(Gender == "F", 1, 0),
         hispanic = ifelse(Ethnicity %in% c("White", "Asian", "Native America"), 1, 0))

# lmer - for alpha metrics
# unconditional model
fit <- lmer(Observed ~ 1 + (1 | SubjectID),
            data = mydata)
summary(fit)
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: Observed ~ 1 + (1 | SubjectID)
   Data: mydata

REML criterion at convergence: 276.8

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.8526 -0.4025  0.1403  0.4613  2.4743 

Random effects:
 Groups    Name        Variance Std.Dev.
 SubjectID (Intercept) 343.28   18.528  
 Residual               48.25    6.946  
Number of obs: 37, groups:  SubjectID, 11

Fixed effects:
            Estimate Std. Error     df t value Pr(>|t|)    
(Intercept)   80.859      5.722 10.126   14.13  5.4e-08 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
plot(fit)

ICC(fit)
[1] 0.8767751

Fixed Effect of Time

fit <- lmer(Observed ~ 1 + time + (1 | SubjectID),
            data = mydata)
summary(fit)
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: Observed ~ 1 + time + (1 | SubjectID)
   Data: mydata

REML criterion at convergence: 274.4

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.8765 -0.3031  0.1047  0.4305  2.4975 

Random effects:
 Groups    Name        Variance Std.Dev.
 SubjectID (Intercept) 343.95   18.546  
 Residual               49.22    7.016  
Number of obs: 37, groups:  SubjectID, 11

Fixed effects:
            Estimate Std. Error      df t value Pr(>|t|)    
(Intercept)  79.9744     5.8781 11.1992  13.606 2.59e-08 ***
time          0.7044     1.0440 25.4952   0.675    0.506    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
     (Intr)
time -0.223
plot(fit)

# plot
dat <- cbind(mydata, fit=predict(fit))
ggplot(dat, aes(time, Observed, group=SubjectID))+
  geom_line(aes(y=fit))+
  geom_point(alpha=0.5)

Fixed and Random effect of Time

fit <- lmer(Observed ~ 1 + time + (1 + time || SubjectID),
            data = mydata)
summary(fit)
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: Observed ~ 1 + time + (1 + time || SubjectID)
   Data: mydata

REML criterion at convergence: 274.4

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.8761 -0.3024  0.1030  0.4303  2.4973 

Random effects:
 Groups      Name        Variance  Std.Dev.
 SubjectID   (Intercept) 343.93536 18.5455 
 SubjectID.1 time          0.03332  0.1825 
 Residual                 49.17013  7.0121 
Number of obs: 37, groups:  SubjectID, 11

Fixed effects:
            Estimate Std. Error      df t value Pr(>|t|)    
(Intercept)  79.9745     5.8777 10.8967  13.606 3.51e-08 ***
time          0.7044     1.0451  9.6234   0.674    0.516    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
     (Intr)
time -0.222
ranova(fit) # can take out random effect time
ANOVA-like table for random-effects: Single term deletions

Model:
Observed ~ time + (1 | SubjectID) + (0 + time | SubjectID)
                               npar  logLik    AIC    LRT Df Pr(>Chisq)    
<none>                            5 -137.22 284.44                         
(1 | SubjectID)                   4 -153.74 315.47 33.035  1   9.05e-09 ***
time in (0 + time | SubjectID)    4 -137.22 282.44  0.000  1     0.9965    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
plot(fit)

dat <- cbind(mydata, fit=predict(fit))

ggplot(dat, aes(time, Observed, group=SubjectID))+
  geom_line(aes(y=fit))+
  geom_point(alpha=0.5)#+

  #geom_abline(intercept = fixef(fit)[1], slope=fixef(fit)[2],
              #linetype="dashed", size=1.5)

Adding demographics

Time is only a fixed effect.

fit <- lmer(Observed ~ 1 + time + female + hispanic + (1 | SubjectID),
            data = mydata)
summary(fit)
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: Observed ~ 1 + time + female + hispanic + (1 | SubjectID)
   Data: mydata

REML criterion at convergence: 260.1

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.8378 -0.2979  0.1045  0.3988  2.5353 

Random effects:
 Groups    Name        Variance Std.Dev.
 SubjectID (Intercept) 404.90   20.122  
 Residual               49.24    7.017  
Number of obs: 37, groups:  SubjectID, 11

Fixed effects:
            Estimate Std. Error      df t value Pr(>|t|)    
(Intercept)  73.7523    12.2582  8.2396   6.017 0.000283 ***
time          0.7036     1.0451 25.3935   0.673 0.506875    
female        9.4770    13.1744  8.0765   0.719 0.492215    
hispanic      0.3278    13.1588  8.0383   0.025 0.980733    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
         (Intr) time   female
time     -0.118              
female   -0.537 -0.011       
hispanic -0.537  0.028 -0.213
ranova(fit)
ANOVA-like table for random-effects: Single term deletions

Model:
Observed ~ time + female + hispanic + (1 | SubjectID)
                npar  logLik    AIC    LRT Df Pr(>Chisq)    
<none>             6 -130.03 272.06                         
(1 | SubjectID)    5 -151.64 313.28 43.216  1  4.902e-11 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
plot(fit)

ICC(fit)
[1] 0.8915758
dat <- cbind(mydata, fit=predict(fit))

ggplot(dat, aes(time, Observed, group=SubjectID, color=Gender))+
  geom_line(aes(y=fit))+
  geom_point(alpha=0.5)

Intervention effect

Overall Effect

fit <- lmer(Observed ~ 1 + intB + female + hispanic + (1 | SubjectID),
            data = mydata)
summary(fit)
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: Observed ~ 1 + intB + female + hispanic + (1 | SubjectID)
   Data: mydata

REML criterion at convergence: 255.3

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.8214 -0.4218  0.1150  0.4418  2.5037 

Random effects:
 Groups    Name        Variance Std.Dev.
 SubjectID (Intercept) 455.39   21.340  
 Residual               48.28    6.948  
Number of obs: 37, groups:  SubjectID, 11

Fixed effects:
            Estimate Std. Error     df t value Pr(>|t|)   
(Intercept)   73.151     13.678  7.089   5.348  0.00102 **
intB           4.797     14.012  7.130   0.342  0.74198   
female        10.329     14.106  7.115   0.732  0.48746   
hispanic      -1.607     14.767  7.046  -0.109  0.91639   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
         (Intr) intB   female
intB     -0.337              
female   -0.557  0.155       
hispanic -0.365 -0.334 -0.250
plot(fit)

ICC(fit)
[1] 0.9041487
dat <- cbind(mydata, fit=predict(fit))

ggplot(dat, aes(time, Observed, group=SubjectID, color=Intervention))+
  geom_line(aes(y=fit))+
  geom_point(alpha=0.5)

With Time

fit <- lmer(Observed ~ 1 + intB*time + (1 | SubjectID),
            data = mydata)
summary(fit)
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: Observed ~ 1 + intB * time + (1 | SubjectID)
   Data: mydata

REML criterion at convergence: 263.1

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-2.79078 -0.27419  0.04933  0.40138  2.41053 

Random effects:
 Groups    Name        Variance Std.Dev.
 SubjectID (Intercept) 374.98   19.364  
 Residual               49.09    7.006  
Number of obs: 37, groups:  SubjectID, 11

Fixed effects:
            Estimate Std. Error     df t value Pr(>|t|)    
(Intercept)   77.063      8.267  9.975   9.322 3.07e-06 ***
intB           6.391     12.279 10.030   0.520    0.614    
time           1.733      1.403 24.342   1.235    0.229    
intB:time     -2.291      2.097 24.504  -1.093    0.285    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
          (Intr) intB   time  
intB      -0.673              
time      -0.218  0.146       
intB:time  0.146 -0.213 -0.669
ranova(fit)
ANOVA-like table for random-effects: Single term deletions

Model:
Observed ~ intB + time + (1 | SubjectID) + intB:time
                npar  logLik    AIC    LRT Df Pr(>Chisq)    
<none>             6 -131.53 275.06                         
(1 | SubjectID)    5 -152.63 315.27 42.206  1  8.215e-11 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
plot(fit)

ICC(fit)
[1] 0.8842445
dat <- cbind(mydata, fit=predict(fit))

ggplot(dat, aes(time, Observed, group=SubjectID, color=Intervention))+
  geom_line(aes(y=fit))+
  geom_point(alpha=0.5)

Alpha: Shannon Diversity

Unconditional Model

# lmer - for alpha metrics
# unconditional model
fit <- lmer(Shannon ~ 1 + (1 | SubjectID),
            data = mydata)
summary(fit)
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: Shannon ~ 1 + (1 | SubjectID)
   Data: mydata

REML criterion at convergence: 27.5

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-2.52108 -0.10901  0.09227  0.42858  2.24145 

Random effects:
 Groups    Name        Variance Std.Dev.
 SubjectID (Intercept) 0.20910  0.4573  
 Residual              0.05614  0.2369  
Number of obs: 37, groups:  SubjectID, 11

Fixed effects:
            Estimate Std. Error      df t value Pr(>|t|)    
(Intercept)   2.5369     0.1441 10.0567    17.6 6.93e-09 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
plot(fit)

ICC(fit)
[1] 0.7883436

Fixed Effect of Time

fit <- lmer(Shannon ~ 1 + time + (1 | SubjectID),
            data = mydata)
summary(fit)
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: Shannon ~ 1 + time + (1 | SubjectID)
   Data: mydata

REML criterion at convergence: 31.8

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-2.56270 -0.23068  0.06454  0.43411  2.28067 

Random effects:
 Groups    Name        Variance Std.Dev.
 SubjectID (Intercept) 0.21095  0.4593  
 Residual              0.05688  0.2385  
Number of obs: 37, groups:  SubjectID, 11

Fixed effects:
            Estimate Std. Error       df t value Pr(>|t|)    
(Intercept)  2.50312    0.15156 12.01774  16.516 1.26e-09 ***
time         0.02671    0.03539 25.67099   0.755    0.457    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
     (Intr)
time -0.295
plot(fit)

# plot
dat <- cbind(mydata, fit=predict(fit))
ggplot(dat, aes(time, Shannon, group=SubjectID))+
  geom_line(aes(y=fit))+
  geom_point(alpha=0.5)

Fixed and Random effect of Time

fit <- lmer(Shannon ~ 1 + time + (1 + time || SubjectID),
            data = mydata)
boundary (singular) fit: see ?isSingular
summary(fit)
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: Shannon ~ 1 + time + (1 + time || SubjectID)
   Data: mydata

REML criterion at convergence: 31.8

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-2.56270 -0.23068  0.06454  0.43411  2.28067 

Random effects:
 Groups      Name        Variance Std.Dev.
 SubjectID   (Intercept) 0.21095  0.4593  
 SubjectID.1 time        0.00000  0.0000  
 Residual                0.05688  0.2385  
Number of obs: 37, groups:  SubjectID, 11

Fixed effects:
            Estimate Std. Error       df t value Pr(>|t|)    
(Intercept)  2.50312    0.15156 12.01773  16.516 1.26e-09 ***
time         0.02671    0.03539 25.67099   0.755    0.457    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
     (Intr)
time -0.295
convergence code: 0
boundary (singular) fit: see ?isSingular
ranova(fit) # can take out random effect time
ANOVA-like table for random-effects: Single term deletions

Model:
Shannon ~ time + (1 | SubjectID) + (0 + time | SubjectID)
                               npar  logLik    AIC    LRT Df Pr(>Chisq)    
<none>                            5 -15.904 41.808                         
(1 | SubjectID)                   4 -28.136 64.272 24.464  1  7.573e-07 ***
time in (0 + time | SubjectID)    4 -15.904 39.808  0.000  1          1    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
plot(fit)

dat <- cbind(mydata, fit=predict(fit))

ggplot(dat, aes(time, Shannon, group=SubjectID))+
  geom_line(aes(y=fit))+
  geom_point(alpha=0.5)#+

  #geom_abline(intercept = fixef(fit)[1], slope=fixef(fit)[2],
              #linetype="dashed", size=1.5)

Adding demographics

Time is only a fixed effect.

fit <- lmer(Shannon ~ 1 + time + female + hispanic + (1 | SubjectID),
            data = mydata)
summary(fit)
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: Shannon ~ 1 + time + female + hispanic + (1 | SubjectID)
   Data: mydata

REML criterion at convergence: 32.3

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-2.52044 -0.19196  0.07653  0.43541  2.32382 

Random effects:
 Groups    Name        Variance Std.Dev.
 SubjectID (Intercept) 0.25601  0.5060  
 Residual              0.05687  0.2385  
Number of obs: 37, groups:  SubjectID, 11

Fixed effects:
            Estimate Std. Error       df t value Pr(>|t|)    
(Intercept)  2.39623    0.31436  8.28622   7.623 5.11e-05 ***
time         0.02675    0.03545 25.53005   0.755    0.457    
female       0.20648    0.33656  7.99579   0.613    0.557    
hispanic    -0.03743    0.33585  7.92920  -0.111    0.914    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
         (Intr) time   female
time     -0.156              
female   -0.535 -0.015       
hispanic -0.535  0.036 -0.212
ranova(fit)
ANOVA-like table for random-effects: Single term deletions

Model:
Shannon ~ time + female + hispanic + (1 | SubjectID)
                npar  logLik    AIC    LRT Df Pr(>Chisq)    
<none>             6 -16.157 44.313                         
(1 | SubjectID)    5 -30.816 71.632 29.319  1  6.138e-08 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
plot(fit)

ICC(fit)
[1] 0.8182491
dat <- cbind(mydata, fit=predict(fit))

ggplot(dat, aes(time, Shannon, group=SubjectID, color=Gender))+
  geom_line(aes(y=fit))+
  geom_point(alpha=0.5)

Intervention effect

fit <- lmer(Shannon ~ 1 + intB + time + female +  (1 | SubjectID),
            data = mydata)
summary(fit)
Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: Shannon ~ 1 + intB + time + female + (1 | SubjectID)
   Data: mydata

REML criterion at convergence: 32.1

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.5421 -0.2129  0.0820  0.4446  2.3060 

Random effects:
 Groups    Name        Variance Std.Dev.
 SubjectID (Intercept) 0.24560  0.4956  
 Residual              0.05679  0.2383  
Number of obs: 37, groups:  SubjectID, 11

Fixed effects:
            Estimate Std. Error       df t value Pr(>|t|)    
(Intercept)  2.28187    0.30569  8.58746   7.465 4.93e-05 ***
intB         0.18778    0.31332  8.16494   0.599    0.565    
time         0.02692    0.03539 25.65615   0.761    0.454    
female       0.21545    0.32387  8.13434   0.665    0.524    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
       (Intr) intB   time  
intB   -0.522              
time   -0.143  0.005       
female -0.713  0.088 -0.007
ranova(fit)
ANOVA-like table for random-effects: Single term deletions

Model:
Shannon ~ intB + time + female + (1 | SubjectID)
                npar  logLik    AIC    LRT Df Pr(>Chisq)    
<none>             6 -16.037 44.073                         
(1 | SubjectID)    5 -30.640 71.280 29.207  1  6.503e-08 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
plot(fit)

ICC(fit)
[1] 0.8121915
dat <- cbind(mydata, fit=predict(fit))

ggplot(dat, aes(time, Shannon, group=SubjectID, color=Intervention))+
  geom_line(aes(y=fit))+
  geom_point(alpha=0.5)


sessionInfo()
R version 3.6.1 (2019-07-05)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 18362)

Matrix products: default

locale:
[1] LC_COLLATE=English_United States.1252 
[2] LC_CTYPE=English_United States.1252   
[3] LC_MONETARY=English_United States.1252
[4] LC_NUMERIC=C                          
[5] LC_TIME=English_United States.1252    

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] viridis_0.5.1     viridisLite_0.3.0 gridExtra_2.3     xtable_1.8-4     
 [5] kableExtra_1.1.0  plyr_1.8.4        data.table_1.12.6 readxl_1.3.1     
 [9] forcats_0.4.0     stringr_1.4.0     dplyr_0.8.3       purrr_0.3.3      
[13] readr_1.3.1       tidyr_1.0.0       tibble_2.1.3      ggplot2_3.2.1    
[17] tidyverse_1.3.0   lmerTest_3.1-1    lme4_1.1-21       Matrix_1.2-17    
[21] phyloseq_1.30.0  

loaded via a namespace (and not attached):
 [1] nlme_3.1-140        fs_1.3.1            lubridate_1.7.4    
 [4] RColorBrewer_1.1-2  webshot_0.5.2       httr_1.4.1         
 [7] rprojroot_1.3-2     numDeriv_2016.8-1.1 tools_3.6.1        
[10] backports_1.1.5     R6_2.4.1            vegan_2.5-6        
[13] DBI_1.0.0           lazyeval_0.2.2      BiocGenerics_0.32.0
[16] mgcv_1.8-28         colorspace_1.4-1    permute_0.9-5      
[19] ade4_1.7-13         withr_2.1.2         tidyselect_0.2.5   
[22] compiler_3.6.1      git2r_0.26.1        cli_1.1.0          
[25] rvest_0.3.5         Biobase_2.46.0      xml2_1.2.2         
[28] labeling_0.3        scales_1.1.0        digest_0.6.23      
[31] minqa_1.2.4         rmarkdown_1.18      XVector_0.26.0     
[34] pkgconfig_2.0.3     htmltools_0.4.0     dbplyr_1.4.2       
[37] rlang_0.4.2         rstudioapi_0.10     farver_2.0.1       
[40] generics_0.0.2      jsonlite_1.6        magrittr_1.5       
[43] biomformat_1.14.0   Rcpp_1.0.3          munsell_0.5.0      
[46] S4Vectors_0.24.1    Rhdf5lib_1.8.0      ape_5.3            
[49] lifecycle_0.1.0     stringi_1.4.3       yaml_2.2.0         
[52] MASS_7.3-51.4       zlibbioc_1.32.0     rhdf5_2.30.1       
[55] grid_3.6.1          parallel_3.6.1      promises_1.1.0     
[58] crayon_1.3.4        lattice_0.20-38     Biostrings_2.54.0  
[61] haven_2.2.0         splines_3.6.1       multtest_2.42.0    
[64] hms_0.5.2           zeallot_0.1.0       knitr_1.26         
[67] pillar_1.4.2        igraph_1.2.4.2      boot_1.3-22        
[70] reshape2_1.4.3      codetools_0.2-16    stats4_3.6.1       
[73] reprex_0.3.0        glue_1.3.1          evaluate_0.14      
[76] modelr_0.1.5        vctrs_0.2.0         nloptr_1.2.1       
[79] httpuv_1.5.2        foreach_1.4.7       cellranger_1.1.0   
[82] gtable_0.3.0        assertthat_0.2.1    xfun_0.11          
[85] broom_0.5.2         later_1.0.0         survival_2.44-1.1  
[88] iterators_1.0.12    IRanges_2.20.1      workflowr_1.5.0    
[91] cluster_2.1.0