# Introduction to hypothesis testing for diversity

#### Amy Willis

#### 2022-11-21

This tutorial will talk through hypothesis testing for alpha diversity indices using the functions `betta` and `betta_random`.

## Disclaimer

Disclaimer: If you have not taken a introductory statistics class or devoted serious time to learning introductory statistics, I strongly encourage you to reconsider doing so before ever quoting a p-value or doing modeling of any kind. An introductory statistics class will teach you valuable skills that will serve you well throughout your entire scientific career, including the use and abuse of p-values in science, how to responsibly fit models and test null hypotheses, and an appreciation for how easy it is to inflate the statistical significance of a result. Please equip yourself with the statistical skills and scepticism necessary to responsibly test and discuss null hypothesis significance testing.

## Preliminaries

Download the latest version of the package from github.

Let’s use the Whitman et al dataset as our example.

    data("soil_phylo")
    soil_phylo %>% sample_data %>% head

    ##      Plants DayAmdmt Amdmt ID Day
    ## S009      1       01     1  D   0
    ## S204      1       21     1  D   2
    ## S112      0       11     1  B   1
    ## S247      0       22     2  F   2
    ## S026      0       00     0  A   0
    ## S023      1       00     0  C   0

I’m only going to consider samples amended with biochar, and I want to look at the effect of `Day`. This will tell us how much diversity in the soil changed from Day 0 to Day 82. (Just to be confusing, Day 82 is called Day 2 in the dataset.)

    subset_soil <- soil_phylo %>%
      subset_samples(Amdmt == 1) %>% # only biochar
      subset_samples(Day %in% c(0, 2))  # only Days 0 and 82

I now run breakaway on these samples to get richness estimates, and plot them.

    richness_soil <- subset_soil %>% breakaway
    plot(richness_soil, physeq=subset_soil, color="Day", shape = "ID")

Don’t freak out! Those are wide error bars, but nothing went wrong – it’s just really hard to estimate the true number of unknown species in soil. `breakaway` was developed to deal with this, and to make sure that we account for that uncertainty when we do inference.

We can get a table of the estimates and their uncertainties as follows:

    summary(richness_soil) %>% tibble::as_tibble()

    ## # A tibble: 32 × 7
    ##    estimate  error lower   upper sample_names name      model
    ##       <dbl>  <dbl> <dbl>   <dbl> <chr>        <chr>     <chr>
    ##  1    5257.  203.  4010.  54003. S009         breakaway Kemp
    ##  2    5343. 2252.  4317. 316529. S204         breakaway Kemp
    ##  3    4609.  978.  2782. 250548. S012         breakaway Kemp
    ##  4    5446. 1393.  3864. 299572. S207         breakaway Kemp
    ##  5    5359.  143.  4370.  35608. S202         breakaway Kemp
    ##  6    3882.  181.  2802.  44399. S007         breakaway Kemp
    ##  7    4906. 4787.  3111. 748118. S022         breakaway Kemp
    ##  8    5215.  343.  3327. 106472. S024         breakaway Negative Binomial
    ##  9    4509.   98.9 3443.  24952. S032         breakaway Kemp
    ## 10    5070.  114.  4189.  27300. S212         breakaway Kemp
    ## # … with 22 more rows

If you haven’t seen a `tibble` before, it’s like a `data.frame`, but way better. Already we can see that we only have 10 rows printed as opposed to the usual bagillion.

## Inference

The first step to doing inference is to decide on your design matrix. We need to grab our covariates into a data frame (or tibble), so let’s start by doing that:

    meta <- subset_soil %>%
      sample_data %>%
      tibble::as_tibble() %>%
      dplyr::mutate("sample_names" = subset_soil %>% sample_names )

That warning is not a problem – it’s just telling us that it’s not a phyloseq object anymore.

Suppose we want to fit the model with Day as a fixed effect. Here’s how we do that,

    combined_richness <- meta %>%
      dplyr::left_join(summary(richness_soil),
                by = "sample_names")
    # Old way (still works)
    bt_day_fixed <- betta(chats = combined_richness$estimate,
                          ses = combined_richness$error,
                          X = model.matrix(~Day, data = combined_richness))
    # Fancy new way -- thanks to Sarah Teichman for implementing!
    bt_day_fixed <- betta(formula = estimate ~ Day,
                          ses = error, data = combined_richness)
    bt_day_fixed$table

    ##             Estimates Standard Errors p-values
    ## (Intercept) 4547.1078        125.8542    0.000
    ## Day2         139.5325        170.1855    0.412

So we see an estimated increase in richness after 82 days of 122 taxa, with the standard error in the estimate of 171. A hypothesis test for a change in richness (i.e., testing a null hypothesis of no change) would not be rejected at any reasonable cut-off (p = 0.412).

Alternatively, we could fit the model with plot ID as a random effect. Here’s how we do that:

    # Old way (still works)
    bt_day_fixed_id_random <- betta_random(chats = combined_richness$estimate,
                                           ses = combined_richness$error,
                                           X = model.matrix(~Day, data = combined_richness),
                                           groups=combined_richness$ID)
    # Fancy new way
    bt_day_fixed_id_random <-
      betta_random(formula = estimate ~ Day | ID,
                   ses = error,  data = combined_richness)
    bt_day_fixed_id_random$table

    ##             Estimates Standard Errors p-values
    ## (Intercept) 4475.4249        119.2010     0.00
    ## Day2         257.7939        161.2601     0.11

Under this different model, we see an estimated increase in richness after 82 days of 258 taxa, with the standard error in the estimate of 161. A hypothesis test for a change in richness still would not be rejected at any reasonable cut-off (p = 0.11).

If you choose to use the old way, the structure of `betta_random` is to input your design matrix as `X`, and your random effects as `groups`, where the latter is a categorical variable. Otherwise, the input looks like how you would hand this off to a regular mixed effects model in the package `lme4`!

If you are interested in generating confidence intervals for and testing hypotheses about linear combinations of fixed effects estimated in a `betta` or `betta_random` model, we recommend using the `betta_lincom` function.

For example, to generate a confidence interval for \(\beta_0 + \beta_1\) (i.e., intercept plus ‘Day2’ coefficient, or in other words, the mean richness in soils on day 82 of the experiment) using the model we fit in the previous code chunk, we run the following code:

    betta_lincom(fitted_betta = bt_day_fixed_id_random,
                 linear_com = c(1,1),
                 signif_cutoff = 0.05)

    ##   Estimates Standard Errors Lower CIs Upper CIs  p-values
    ## 1  4733.219        161.2601  4417.155  5049.283 1.14e-189

Here, we’ve set the `linear_com` argument equal to `c(1,1)` to tell `betta_lincom` to construct a confidence interval for \(1 \times \beta_0 + 1 \times \beta_1\). Because we set `signif_cutoff` equal to 0.05, `betta_lincom` returns a \(95\% = (1 - 0.05)*100\%\) confidence interval. The p-value reported here is for a test of the null hypothesis that \(1 \times \beta_0 + 1 \times \beta_1 = 0\) – unsurprisingly, this is small. (If you are confused about why this is “unsurprising,” remember that \(\beta_0 + \beta_1\) represents a mean richness in soils on day 82 of the experiment of Whitman et al. When can richness be zero?)

The syntax and output using `betta_lincom` with a `betta` object as input is exactly the same as with a `betta_random` object, so we haven’t included a separate example for this case.

To look at a more complicated example of hypothesis testing, let’s now include another date of observation in the Whitman et al. dataset – `Day = 1`, or observations taken on day 12 of this study. We might be interested now in determining whether there is any difference across observation times in richness.

We prepare data and fit a model essentially as we did above. First, we subset the soil data to only biochar-amended plots and allow `Day` to equal 0, 1, or 2.

    subset_soil_days_1_2 <- soil_phylo %>%
      subset_samples(Amdmt == 1) %>% # only biochar
      subset_samples(Day %in% c(0, 1, 2))  # Days 0, 12, and 82

We extract metadata and aggregate to phylum level as above as well:

    meta_days_1_2 <- subset_soil_days_1_2 %>%
      sample_data %>%
      tibble::as_tibble() %>%
      dplyr::mutate("sample_names" = subset_soil_days_1_2 %>% sample_names )

We again run DivNet and extract estimates of Shannon diversity.

    richness_days_1_2 <- subset_soil_days_1_2 %>%
      breakaway

    combined_richness_days_1_2 <- meta_days_1_2 %>%
      dplyr::left_join(summary(richness_days_1_2),
                by = "sample_names")
    combined_richness_days_1_2

    ## # A tibble: 48 × 12
    ##    Plants DayAmdmt Amdmt ID    Day   sample_n…¹ estim…² error lower  upper name
    ##    <chr>  <chr>    <chr> <chr> <chr> <chr>        <dbl> <dbl> <dbl>  <dbl> <chr>
    ##  1 1      01       1     D     0     S009         5257.  203. 4010. 5.40e4 brea…
    ##  2 1      21       1     D     2     S204         5343. 2252. 4317. 3.17e5 brea…
    ##  3 0      11       1     B     1     S112         5165. 4017. 3419. 6.57e5 brea…
    ##  4 0      01       1     B     0     S012         4609.  978. 2782. 2.51e5 brea…
    ##  5 1      11       1     D     1     S134         5498.  212. 4166. 5.82e4 brea…
    ##  6 0      21       1     B     2     S207         5446. 1393. 3864. 3.00e5 brea…
    ##  7 0      21       1     B     2     S202         5359.  143. 4370. 3.56e4 brea…
    ##  8 0      01       1     B     0     S007         3882.  181. 2802. 4.44e4 brea…
    ##  9 1      11       1     D     1     S139         5137.  251. 4007. 6.26e4 brea…
    ## 10 0      11       1     B     1     S122         5755. 4281. 4331. 5.97e5 brea…
    ## # … with 38 more rows, 1 more variable: model <chr>, and abbreviated variable
    ## #   names ¹​sample_names, ²​estimate

Now we fit another model with `betta_random`.

    bt_day_1_2_fixed_id_random <- betta_random(formula = estimate ~ Day | ID,
                   ses = error,  data = combined_richness_days_1_2)
    bt_day_1_2_fixed_id_random$table

    ##             Estimates Standard Errors p-values
    ## (Intercept) 4444.5046        92.85784    0.000
    ## Day1         468.4976       178.68946    0.009
    ## Day2         304.8528       146.93937    0.038

The output we get from `betta_random` gives us p-values for testing whether mean richness is the same at day 12 as at day 0 and for whether it is the same at day 82 as at day 0, but we want to get a single p-value for an overall test of whether mean Shannon diversity varies with day at all! To do this, we can use the `test_submodel` function to test our full model against a null with no terms in `Day` using a parametric bootstrap:

    set.seed(345)
    submodel_test <- test_submodel(bt_day_1_2_fixed_id_random,
                              submodel_formula = estimate~1,
                              method = "bootstrap",
                              nboot = 100)

    submodel_test$pval

    ## [1] 0.12

This returns a p-value of 0.12, which means that the null hypothesis would not be rejected at a cut-off of 0.05. This means that we do not have strong enough evidence to reject some difference in mean richness over time. In practice, it’s a good idea to use more than 100 bootstrap iterations – 10,000 is a good choice for publication. (We use 100 here so the vignette loads in a reasonable amount of time.)

And there you have it! That’s how to do hypothesis testing for diversity!

If you use our tools, please don’t forget to cite them!

- `breakaway`: Willis & Bunge. (2015). Estimating diversity via frequency ratios. Biometrics. doi: 10.1111/biom.12332.
- `DivNet`: Willis & Martin. (2020). DivNet: Estimating diversity in networked communities. Biostatistics. doi: 10.1093/biostatistics/kxaa015.
- `betta`: Willis, Bunge & Whitman. (2016). Improved detection of changes in species richness in high diversity microbial communities. Journal of the Royal Statistical Society: Series C. doi: 10.1111/rssc.12206.