# Statistical testing with myTAI

## The motivation for statistically-informed analysis of TAI patterns

Let’s say you have your transcriptome age index (TAI) pattern, e.g.

    library(myTAI)

    data("example_phyex_set_old")
    myTAI::plot_signature(example_phyex_set_old, show_p_val = FALSE)

But how can you test if this pattern is significant?

## Statistical analysis of TAI patterns

With `myTAIv2`, we have introduced a suite of permutation-based statistical testing framework to test the significance of your TAI (as well as TDI, TSI and analogous) profiles.

In this section, we will go through:
(1) `myTAI::stat_flatline_test()` which evaluates whether the observed TAI profile is significantly different from a flatline (i.e. no evolutionary trend).
(2) `myTAI::stat_reductive_hourglass_test()` which evaluates whether the observed TAI profile is significantly different from an hourglass pattern (i.e. consistent with the molecular hourglass hypothesis), as well as the reverse `myTAI::stat_reverse_hourglass_test()` and tests of potential other evolutionary patterns such as `myTAI::stat_early_conservation_test()` and `myTAI::stat_late_conservation_test()`.
(3) `myTAI::stat_pairwise_test()` which evaluates whether the TAI profiles of two groups of samples are significantly different from each other.

Unlike `stat_flatline_test()`, the other tests (`stat_reductive_hourglass_test()`, `stat_reverse_hourglass_test()`, `stat_early_conservation_test()`, `stat_late_conservation_test()`, and `stat_pairwise_test()`) require an a priori grouping of developmental stage or experimental conditions.

## Flatline test

The flat line test tests the variance of the TAI (or equivalent) profile against a null distribution.

    myTAI::stat_flatline_test(example_phyex_set_old, plot_result = TRUE)

Show output

    ## Computing: [========================================] 100% (done)

    ##
    ## Statistical Test Result
    ## =======================
    ## Method: Flat Line Test
    ## Test statistic: 0.03103926
    ## P-value: 3.701579e-07
    ## Alternative hypothesis: greater
    ## Data: Embryogenesis 2011

We can see a histogram of the null distribution (along with the fitted null distribution) and the observed variance. In text, we can see the summary test statistics of the flat line test. The plots are created by default but you can also turn `plot_result = FALSE` to not plot the results of the test.

In this example, we can see that the pattern significantly deviates from a flat line (i.e. no variance).

The `myTAI::stat_flatline_test()` is a good first step to evaluating the evolutionary signals in your transcriptomic data.

### Cullen and Frey plot

To further inspect the permutation test, we can use the Cullen and Frey plot (skewness vs. kurtosis; `myTAI::plot_cullen_frey()`) to help identify appropriate distribution families for the null sample data.

    res_flt <- myTAI::stat_flatline_test(example_phyex_set_old, plot_result = FALSE)

    myTAI::plot_cullen_frey(res_flt)

Show output

    myTAI::plot_cullen_frey(res_flt)

    ## summary statistics
    ## ------
    ## min:  0.0003112649   max:  0.02377656
    ## median:  0.002407031
    ## mean:  0.004064448
    ## estimated sd:  0.004705971
    ## estimated skewness:  2.509826
    ## estimated kurtosis:  9.799639

We can see that the null sample lies within the gamma distribution family, which is a good sign because the `myTAI::stat_flatline_test()` assumes that the null sample is gamma-distributed.

The `myTAI::plot_cullen_frey()` function is also useful for the other tests as well (e.g. `stat_reductive_hourglass_test()`, `stat_reverse_hourglass_test()`, `stat_early_conservation_test()`, `stat_late_conservation_test()`, and `stat_pairwise_test()`).

### Null distribution visualisation

Furthermore, using the results of the flat line test (saved here as `res_flt`), we can visualise TAI pattern of each of the permutations using `myTAI::plot_null_txi_sample()`, i.e.

    myTAI::plot_null_txi_sample(res_flt)

Show output

    myTAI::plot_null_txi_sample(res_flt)

The null variance distribution is produced from the grey lines, while the red line is the observed TAI profile. The distance between the red and grey lines indicate that older genes are generally higher expressed than younger genes (as they contribute more to the TAI), and the observed TAI profile is more variable than the null distribution.

## Reductive hourglass test and other time-course tests

The following tests (`stat_reductive_hourglass_test()`, `stat_reverse_hourglass_test()`, `stat_early_conservation_test()` and `stat_late_conservation_test()`) evaluates changes in evolutionary signal across developmental stages. Rather than testing the variance of the TAI profile against a null distribution, these tests evaluate whether the observed TAI profile is significantly different from the null to support an hourglass pattern (i.e. consistent with the molecular hourglass hypothesis), a reverse hourglass pattern, or an early/late conservation pattern.

We also need to specify a priori grouping of developmental stages or experimental conditions. For example, we can group the developmental stages into: early (stages 1-3), mid (stages 4-5) and late (stages 6-7). This is specified via the parameter `modules`.

### Reductive hourglass test

The reductive hourglass test evaluates whether the observed TAI profile significantly deviates from the null towards an `hourglass` pattern, i.e. the TAI values are lower in the mid-stage compared to the early and late stages. This can be achieved with the function `myTAI::stat_reductive_hourglass_test()`.

    modules <- list(early = 1:3, mid = 4:5, late = 6:7)
    myTAI::stat_reductive_hourglass_test(
      example_phyex_set_old, plot_result = TRUE,
      modules = modules)

Show output

    ##
    ## Statistical Test Result
    ## =======================
    ## Method: Reductive Hourglass Test
    ## Test statistic: 0.1111285
    ## P-value: 0.006927359
    ## Alternative hypothesis: greater
    ## Data: Embryogenesis 2011

In this example, we can see that the TAI pattern is consistent with the molecular hourglass model, as the observed TAI profile is significantly different from the null distribution.

Cullen and Frey plot

We can also check the Cullen and Frey plot (though I have put this under a details tag to not clutter the output) to see if the null sample is normally-distributed, which the assumption for the `myTAI::stat_reductive_hourglass_test()`.

    res_flt <- myTAI::stat_reductive_hourglass_test(
      example_phyex_set_old, plot_result = FALSE,
      modules = modules)

    myTAI::plot_cullen_frey(res_flt)

    ## summary statistics
    ## ------
    ## min:  -0.2195237   max:  0.1520561
    ## median:  -0.03085595
    ## mean:  -0.02621749
    ## estimated sd:  0.05637542
    ## estimated skewness:  -0.1419124
    ## estimated kurtosis:  6.185108

Indeed it is!

### Reverse hourglass test

Analogously, we can also test for whether the observed patterns are consistent with an inverse or `reverse hourglass` pattern, i.e. the TAI (as well as TDI, TSI and analogous) values tend to be higher in the mid-stage. This can be achieved with the function `myTAI::stat_reverse_hourglass_test()`.

    modules <- list(early = 1:3, mid = 4:5, late = 6:7)
    myTAI::stat_reverse_hourglass_test(
      example_phyex_set_old, plot_result = TRUE,
      modules = modules)

Show output

    ##
    ## Statistical Test Result
    ## =======================
    ## Method: Reverse Hourglass Test
    ## Test statistic: -0.2985458
    ## P-value: 0.9999908
    ## Alternative hypothesis: greater
    ## Data: Embryogenesis 2011

As we can see from the p-values of the one-tailed test, the TAI profile does not support the reverse hourglass pattern, i.e. the TAI values are not higher in the mid-stage. This is not surprisingly given the results of the `myTAI::stat_reductive_hourglass_test()`.

### Early conservation test

Another potential pattern of so-called “ontogeny-phylogeny” correlation is `early conservation`, which has a long history in evolutionary developmental biology. This pattern suggests that early developmental stages are more conserved across species, while later stages show more variability.

To test for the early conservation pattern, we can use the function `myTAI::stat_early_conservation_test()`. This test evaluates whether the TAI profile is significantly lower in the early stages.

    modules <- list(early = 1:3, mid = 4:5, late = 6:7)
    myTAI::stat_early_conservation_test(
      example_phyex_set_old, plot_result = TRUE,
      modules = modules)

Show output

    ##
    ## Statistical Test Result
    ## =======================
    ## Method: Early Conservation Test
    ## Test statistic: -0.1111285
    ## P-value: 0.8751707
    ## Alternative hypothesis: greater
    ## Data: Embryogenesis 2011

Again, the TAI profile does not support the early conservation pattern, as the p-value is not significant. TAI values are not significantly lower in the early stages compared to the mid and late stages.

### Late conservation test

Another potential pattern is `late conservation`, which suggests that late developmental stages are more conserved across species compared to early stages. To test for the late conservation pattern, we can use the function `myTAI::stat_late_conservation_test()`, i.e.

    modules <- list(early = 1:3, mid = 4:5, late = 6:7)
    myTAI::stat_late_conservation_test(
      example_phyex_set_old, plot_result = TRUE,
      modules = modules)

Show output

    ##
    ## Statistical Test Result
    ## =======================
    ## Method: Late Conservation Test
    ## Test statistic: -0.2985458
    ## P-value: 0.9998352
    ## Alternative hypothesis: greater
    ## Data: Embryogenesis 2011

As expected again, we do not see support for the late conservation pattern, as the TAI values are not significantly lower in the late stages compared to the early and mid stages.

## Pairwise test

Finally, we can also test whether the TAI profiles of two groups of samples are significantly different from each other. This can be achieved with the function `myTAI::stat_pairwise_test()`.

For example, let’s say we have an experiment where we disrupt an organisms’ development at a given stage using a certain perturbation (e.g. drug treatment). We can compare the TAI profiles of the control and treatment groups to see if the perturbation has an effect on the TAI profile. This could be interesting if the perturbed pathway is thought to recruit orphan genes, i.e. genes that are not conserved across species. An example can be found here.

With `myTAIv2`, we can facilitate such test. Since we do not have the data, we will use the example data set `example_phyex_set_old` and create two groups of samples: `control` and `treatment`. We can then compare the TAI profiles of these two groups using the `myTAI::stat_pairwise_test()` function.

    modules <- list(contrast1 = 1:4, contrast2 = 5:7)
    myTAI::stat_pairwise_test(
      example_phyex_set_old, plot_result = TRUE,
      modules = modules,
      alternative = "greater")

Show output

    ##
    ## Statistical Test Result
    ## =======================
    ## Method: Pairwise Test
    ## Test statistic: -0.0942979
    ## P-value: 0.9425973
    ## Alternative hypothesis: greater
    ## Data: Embryogenesis 2011

In this mock example, we can see that the TAI profile of the control group (`contrast1`) is not significantly higher than the treatment group (`contrast2`), indicating that the perturbation has no effect on the TAI profile. The p-value is significant, and the observed TAI profile is significantly different from the null distribution.

## Summary

In this section, we have introduced the statistical testing framework of `myTAIv2` to evaluate the significance of TAI patterns. With this, we can address the question “is my TAI pattern significant?” in a statistically-informed manner.

We have also introduced the Cullen and Frey plot to help identify appropriate distribution families for the null sample data, as well as the null distribution visualisation to inspect the permutation test results.

It should be noted, however, that the results of the tests may be sensitive to the grouping of developmental stages or experimental conditions. This a priori information should be handled with care.

Moreover, RNA-seq data transformation (e.g. log2 transformation) can also affect the results of the tests and the overall TAI profile. Given that we do not have a gold standard transformation, it is recommended to try different transformations (using `myTAI::tf()`) and evaluate the `robustness` of the results. For more information, check out this article.