PRE_POST_STATS.Rmd

---
title: "Statistical analyses of post- vs. pre-training body composition and VO2max testing measures"
author: Tyler Sagendorf
date: "`r format(Sys.time(), '%d %B, %Y')`"
output: 
  rmarkdown::html_document:
    toc: true
vignette: >
  %\VignetteIndexEntry{Statistical analyses of post- vs. pre-training body composition and VO2max testing measures}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 7,
  fig.height = 5,
  warning = FALSE,
  message = FALSE
)
```

```{r setup, message=FALSE, warning=FALSE}
# Required packages
library(MotrpacRatTrainingPhysiologyData)
library(ggplot2)
library(MASS)
library(dplyr)
library(emmeans)
library(tibble)
library(tidyr)
library(purrr)
library(latex2exp)
library(rstatix)
theme_set(theme_bw()) # base plot theme
```


# Regression Models

We will fit linear regression models with age, sex, group, and their interactions as predictors of the differences between post- and pre-training values for different measures of body mass, body composition, and VO$_2$max. If heteroscedasticity is observed, we will include reciprocal group variances as weights. Certain observations may be removed if their inclusion substantially affects the model fit. Model parsimony will be achieved through ANOVA F-tests.


## NMR Body Mass

Body mass recorded on the same day as the NMR body composition measures.

```{r, fig.height=3}
# Plot points
NMR %>% 
  filter(!is.na(post_body_mass)) %>% 
  droplevels.data.frame() %>% 
  ggplot(aes(x = group, y = post_body_mass - pre_body_mass)) +
  geom_point(position = position_jitter(width = 0.15, height = 0),
             na.rm = TRUE, alpha = 0.5) +
  facet_wrap(~ sex + age, nrow = 1) +
  labs(y = "NMR total body mass (post - pre)") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1, vjust = 0.5))
```

There is one negative outlying value in the 6M SED male group that may affect the model fit. We will start with a WLS model, since we observe different group variances.

```{r}
wt.body_mass <- NMR %>% 
  group_by(age, sex, group) %>% 
  mutate(1 / var(post_body_mass - pre_body_mass, na.rm = TRUE)) %>% 
  pull(-1)

fit.body_mass <- lm(I(post_body_mass - pre_body_mass) ~ age * sex * group,
                    weights = wt.body_mass,
                    data = NMR)
plot_lm(fit.body_mass)
```

The diagnostic plots appear fine.

```{r}
anova(fit.body_mass, test = "F")
```


## 1W, 2W: Terminal - Pre NMR Body Mass

Pre: body mass recorded on the same day as the NMR body composition measures in the 1W and 2W groups. Post: body mass recorded upon death (terminal body mass).

```{r, fig.height=3}
term_nmr_df <- BODY_MASSES %>% 
  filter(group %in% c("1W", "2W")) %>% 
  select(pid, iowa_id, age, sex, group, term_body_mass, nmr_pre_body_mass) %>% 
  mutate(diff = term_body_mass - nmr_pre_body_mass) %>% 
  na.omit()

# Plot points
p_term <- ggplot(term_nmr_df, aes(x = group, y = diff)) +
  geom_point(position = position_jitter(width = 0.15, height = 0),
             alpha = 0.5) +
  facet_grid(~ sex + age) +
  labs(y = "terminal - NMR body mass (pre)") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1, vjust = 0.5))

p_term
```

There is an extreme observation in both 6M Male groups.

```{r, fig.height=3, fig.width=4}
ggplot(term_nmr_df, aes(x = nmr_pre_body_mass, y = term_body_mass)) +
  geom_point(na.rm = TRUE, alpha = 0.5) +
  stat_smooth(method = "lm", formula = y ~ x, se = FALSE, na.rm = TRUE)
```

We see the outlier in the 6M 2W male group is an outlier in the scatterplot. We will plot the data without that observation.

```{r, fig.height=3}
# Remove those two points and update the plot
p_term$data <- filter(term_nmr_df, diff > -50)
p_term
```

Nearly every group has at least one outlying sample, and the groups have different variances. We will remove that extremely negative outlier and fit a WLS model.

```{r}
out.term_nmr <- which(term_nmr_df$diff < -50) # 58

wt.term_nmr <- term_nmr_df %>%
  mutate(term_body_mass = ifelse(1:n() %in% out.term_nmr, 
                                 NA, term_body_mass)) %>% 
  group_by(age, sex, group) %>% 
  mutate(1 / var(term_body_mass - nmr_pre_body_mass, na.rm = TRUE)) %>% 
  pull(-1)

fit.term_nmr <- lm(I(term_body_mass - nmr_pre_body_mass) ~ age * sex * group,
                   weights = wt.term_nmr,
                   subset = -out.term_nmr,
                   data = term_nmr_df)
plot_lm(fit.term_nmr)
```

The diagnostic plots look mostly fine.

```{r}
anova(fit.term_nmr, test = "F")
```


## NMR Lean Mass

Lean mass (g) recorded via NMR.

```{r, fig.height=3}
# Plot points
NMR %>% 
  filter(!is.na(post_lean)) %>% 
  droplevels.data.frame() %>% 
  ggplot(aes(x = group, y = post_lean - pre_lean)) +
  geom_point(position = position_jitter(width = 0.15, height = 0),
             na.rm = TRUE, alpha = 0.5) +
  facet_wrap(~ sex + age, nrow = 1) +
  labs(y = "NMR lean mass (post - pre)") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1, vjust = 0.5))
```

We observe several outlying values in several of the 6M male groups.

```{r}
fit.lean <- lm(I(post_lean - pre_lean) ~ age * sex * group,
               data = NMR)
plot_lm(fit.lean)
```

Observations 80, 89, and 140 are outlying, and greatly impact group variances. We will try inverse group variance weights.

```{r}
NMR[c(80, 89, 140), ] %>% 
  select(pid, iowa_id, age, sex, group, pre_lean, post_lean) %>% 
  mutate(diff = post_lean - pre_lean)
```

```{r}
wt.lean <- NMR %>% 
  group_by(age, sex, group) %>% 
  mutate(1 / var(post_lean - pre_lean, na.rm = TRUE)) %>% 
  pull(-1)

fit.lean <- update(fit.lean, weights = wt.lean)
plot_lm(fit.lean)
```

The diagnostic plots seem mostly fine.

```{r}
anova(fit.lean, test = "F")
```


## NMR Fat Mass

Fat mass (g) recorded via NMR.

```{r, fig.height=3}
# Plot points
NMR %>% 
  filter(!is.na(post_fat)) %>% 
  droplevels.data.frame() %>% 
  ggplot(aes(x = group, y = post_fat - pre_fat)) +
  geom_point(position = position_jitter(width = 0.15, height = 0),
             na.rm = TRUE, alpha = 0.5) +
  facet_wrap(~ sex + age, nrow = 1) +
  labs(y = "NMR fat mass (post - pre)") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1, vjust = 0.5))
```

We observe unequal group variances, and several outlying values (especially in the 6M 4W male group). We will fit a WLS model with reciprocal group variances as weights.

```{r}
wt.fat <- NMR %>% 
  group_by(age, sex, group) %>% 
  mutate(1 / var(post_fat - pre_fat, na.rm = TRUE)) %>% 
  pull(-1)

fit.fat <- lm(I(post_fat - pre_fat) ~ age * sex * group,
              weights = wt.fat,
              data = NMR)
plot_lm(fit.fat)
```

The diagnostic plots seem mostly fine.

```{r}
anova(fit.fat, test = "F")
```


## NMR % Lean Mass

Lean mass (g) recorded via NMR divided by the body mass (g) recorded on the same day and then multiplied by 100%.

```{r, fig.height=3}
# Plot points
NMR %>% 
  filter(!is.na(post_lean_pct)) %>% 
  droplevels.data.frame() %>% 
  ggplot(aes(x = group, y = post_lean_pct - pre_lean_pct)) +
  geom_point(position = position_jitter(width = 0.15, height = 0),
             na.rm = TRUE, alpha = 0.5) +
  facet_wrap(~ sex + age, nrow = 1) +
  labs(y = "NMR % lean mass (post - pre)") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1, vjust = 0.5))
```

We observe several outlying values in several of the 6M male groups, similar to the plot of the differences in lean mass. We also observe unequal group variances, so we will fit a WLS model with reciprocal group variances as weights.

```{r}
wt.lean_pct <- NMR %>% 
  group_by(sex, group, age) %>% 
  mutate(1 / var(post_lean_pct - pre_lean_pct, na.rm = TRUE)) %>% 
  pull(-1)

fit.lean_pct <- lm(I(post_lean_pct - pre_lean_pct) ~ age * sex * group,
                   weights = wt.lean_pct,
                   data = NMR)
plot_lm(fit.lean_pct)
```

The diagnostic plots look mostly fine.

```{r}
anova(fit.lean_pct, test = "F")
```


## NMR % Fat Mass

Fat mass (g) recorded via NMR divided by the body mass (g) recorded on the same day and then multiplied by 100%.

```{r, fig.height=3}
# Plot points
NMR %>% 
  filter(!is.na(post_fat_pct)) %>% 
  droplevels.data.frame() %>% 
  ggplot(aes(x = group, y = post_fat_pct - pre_fat_pct)) +
  geom_point(position = position_jitter(width = 0.15, height = 0),
             na.rm = TRUE, alpha = 0.5) +
  facet_wrap(~ sex + age, nrow = 1) +
  labs(y = "NMR % fat mass (post - pre)") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1, vjust = 0.5))
```

There are several groups with outlying values: mainly, two from the 6M 4W males and one from the 18M 8W females. We will try an OLS model first.

```{r}
fit.fat_pct <- lm(I(post_fat_pct - pre_fat_pct) ~ age * sex * group,
                  data = NMR)
plot_lm(fit.fat_pct)
```

Those extreme values are being flagged in the diagnostic plots.

```{r}
NMR[c(136, 138, 218), ] %>% 
  select(pid, iowa_id, age, sex, group, post_fat_pct, pre_fat_pct) %>%
  mutate(diff = post_fat_pct - pre_fat_pct)
```

The two outliers in the 6M 4W male group are about equidistant from the mean of the other values, so their removal will only shrink the variance of the group. Removal of the 18M 8W female sample, however, will reduce the variance and shift the mean away from 0. We will try a WLS model and remove these observations, since they have a huge impact on the group variances and would shift the right tail of the QQ-plot if they were included.

```{r}
# We will remove these values, but keep the rows.
# This is so we can use the subset argument in update, which
# is easier to keep track of when we extract model info later.
wt.fat_pct <- NMR %>% 
  mutate(idx = 1:n(),
         post_fat_pct = ifelse(idx %in% c(136, 138, 218), 
                               NA, post_fat_pct)) %>% 
  group_by(age, sex, group) %>% 
  mutate(1 / var(post_fat_pct - pre_fat_pct, na.rm = TRUE)) %>% 
  pull(-1)

fit.fat_pct <- update(fit.fat_pct, 
                      subset = -c(136, 138, 218), 
                      weights = wt.fat_pct)
plot_lm(fit.fat_pct)
```

The diagnostic plots look fine.

```{r}
anova(fit.fat_pct, test = "F")
```


## Absolute VO$_2$max

Absolute VO$_2$max is calculated by multiplying relative VO$_2$max ($mL \cdot kg^{-1} \cdot min^{-1}$) by body mass (kg).

```{r, fig.height=3}
# Plot points
VO2MAX %>% 
  filter(!is.na(post_vo2max_ml_min)) %>% 
  droplevels.data.frame() %>% 
  ggplot(aes(x = group, y = post_vo2max_ml_min - pre_vo2max_ml_min)) +
  geom_point(position = position_jitter(width = 0.15, height = 0),
             na.rm = TRUE, alpha = 0.5) +
  geom_hline(yintercept = 0, lty = "dashed", color = "red") +
  facet_wrap(~ sex + age, nrow = 1) +
  labs(y = TeX("Absolute VO$_2$max (post - pre)")) +
  theme(axis.text.x = element_text(angle = 90, hjust = 1, vjust = 0.5))
```

Several groups have both large and small outlying values. Namely, 6M SED and 4W males and 18M SED females. Also, we do not have data for the 18M 4W groups of either sex, so certain coefficients will be inestimable. To handle this, we will combine age and group into a single factor.

```{r}
# Concatenate age and group
VO2MAX <- mutate(VO2MAX, age_group = paste0(age, "_", group))

fit.vo2max_abs <- lm(I(post_vo2max_ml_min - pre_vo2max_ml_min) ~ 
                       age_group * sex,
                     data = VO2MAX)
plot_lm(fit.vo2max_abs)
```

The diagnostic plots look mostly fine.

```{r}
anova(fit.vo2max_abs, test = "F")
```


## Relative VO$_2$max

Relative VO$_2$max (mL/kg body mass/min).

```{r, fig.height=3}
# Plot points
VO2MAX %>% 
  filter(!is.na(post_vo2max_ml_kg_min)) %>% 
  droplevels.data.frame() %>% 
  ggplot(aes(x = group, 
             y = post_vo2max_ml_kg_min - pre_vo2max_ml_kg_min)) +
  geom_point(position = position_jitter(width = 0.15, height = 0),
             na.rm = TRUE, alpha = 0.5) +
  geom_hline(yintercept = 0, lty = "dashed", color = "red") +
  facet_grid(~ sex + age) +
  labs(y = "Relative VO2max (post - pre)") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1, vjust = 0.5))
```

There is an outlying sample in the 18M SED female group.

```{r}
fit.vo2max_rel <- lm(I(post_vo2max_ml_kg_min - pre_vo2max_ml_kg_min) ~ 
                       age_group * sex,
                     data = VO2MAX)
plot_lm(fit.vo2max_rel)
```

The diagnostic plots look mostly fine, though we observe unequal group variances. We will include inverse group variance weights in an attempt to stabilize the variance.

```{r}
wt.vo2max_rel <- VO2MAX %>%
  group_by(age_group, sex) %>%
  mutate(1 / var(post_vo2max_ml_kg_min - pre_vo2max_ml_kg_min,
                 na.rm = TRUE)) %>%
  pull(-1)

fit.vo2max_rel <- update(fit.vo2max_rel, weights = wt.vo2max_rel)
plot_lm(fit.vo2max_rel)
```

The diagnostic plots look a bit better, and the mean-variance relationship appears constant.

```{r}
anova(fit.vo2max_rel, test = "F")
```


## Maximum Run Speed

Since run speed was increased in 1.8 m/min increments, as defined in the training protocol, it may be better to use a non-parametric test. Rather than plotting the individual points for the differences, since they can only take on a few different values, we will instead count the number of samples that take on a particular value in each group and scale the points accordingly.

```{r, fig.height=3}
# Plot points
VO2MAX %>% 
  mutate(diff = post_speed_max - pre_speed_max) %>% 
  filter(!is.na(diff)) %>% 
  droplevels.data.frame() %>% 
  count(age, sex, group, diff) %>% 
  ggplot(aes(x = group, y = diff, size = n)) +
  geom_point(na.rm = TRUE) +
  facet_grid(~ age + sex) +
  scale_size_area(max_size = 5) +
  labs(y = "Maximum Run Speed (post - pre)") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1, vjust = 0.5))
```

Most groups only take on 4 or 5 different values. We will use Wilcoxon Rank Sum tests to compare each trained group to their matching control group. We will adjust p-values within each age and sex group after we set up comparisons for all measures.

```{r}
# Testing the differences is equivalent to a paired test
speed_res <- VO2MAX %>% 
  filter(!is.na(post_speed_max)) %>% 
  mutate(speed_diff = post_speed_max - pre_speed_max) %>% 
  group_by(age, sex, group) %>% 
  wilcox_test(formula = speed_diff ~ 1, mu = 0,
              detailed = TRUE,
              p.adjust.method = "none") %>% 
  as.data.frame() %>% 
  dplyr::rename(p.value = p,
                lower.CL = conf.low, 
                upper.CL = conf.high,
                W.ratio = statistic) %>% 
  mutate(across(.cols = where(is.numeric), 
                ~ ifelse(is.nan(.x), NA, .x))) %>% 
  ungroup() %>% 
  dplyr::select(age, sex, group, estimate, lower.CL, upper.CL,
                n, W.ratio, p.value)
```


# Hypothesis Testing

We will test if group means are different from 0 and adjust p-values using the Holm procedure within each combination of sex and age.

```{r}
# We will include the maximum run speed test results at the end
model_list <- list("NMR Body Mass" = fit.body_mass,
                   "Term - NMR Pre Body Mass" = fit.term_nmr,
                   "NMR Lean Mass" = fit.lean,
                   "NMR Fat Mass" = fit.fat,
                   "NMR % Lean" = fit.lean_pct,
                   "NMR % Fat" = fit.fat_pct,
                   "Absolute VO2max" = fit.vo2max_abs,
                   "Relative VO2max" = fit.vo2max_rel)

# Extract model info
model_df <- model_list %>% 
  map_chr(.f = ~ paste(deparse(.x[["call"]]), collapse = "")) %>% 
  enframe(name = "response", 
          value = "model") %>% 
  mutate(model = gsub("(?<=[\\s])\\s*|^\\s+|\\s+$", "", model, perl = TRUE),
         model_type = sub("^([^\\(]+).*", "\\1", model),
         formula = sub(".*formula = ([^,]+),.*", "\\1", model),
         # if weights were used, they were reciprocal group variances
         weights = ifelse(grepl("weights = ", model), 
                          "reciprocal group variances", NA)) %>% 
  dplyr::select(-model) %>% 
  mutate(obs_removed = case_when(
    response == "Term - NMR Pre Body Mass" ~ NMR$iowa_id[58],
    response == "NMR % Fat" ~ paste(NMR$iowa_id[c(136, 138, 218)],
                                    collapse = ", ")
  ))
```

```{r}
# Estimated marginal means
PRE_POST_STATS <- map(model_list, function(mod_i) {
  terms_i <- attr(terms(mod_i), which = "term.labels")
  specs <- intersect(c("age_group", "group"), terms_i)
  by <- intersect(c("age", "sex"), terms_i)
  
  if (length(by) == 0) {
    by <- NULL
  }
  
  out <- emmeans(mod_i, specs = specs, by = by, infer = TRUE) %>% 
    summary() %>% 
    as.data.frame()
  
  if (specs == "age_group") {
    out <- mutate(out,
                  age = sub("_.*", "", age_group),
                  group = sub(".*_", "", age_group),
                  age_group = NULL)
  }
  
  return(out)
}) %>%
  # Include maximum run speed results
  c(list("Maximum Run Speed" = speed_res)) %>% 
  enframe(name = "response") %>% 
  unnest(value) %>%
  mutate(sex = ifelse(is.na(sex), "Female, Male", as.character(sex))) %>% 
  separate_longer_delim(cols = sex, delim = ", ") %>%
  group_by(response, age, sex) %>% 
  mutate(p.adj = p.adjust(p.value, method = "holm")) %>% 
  ungroup() %>%
  mutate(signif = p.adj < 0.05,
         age = factor(age, levels = c("6M", "18M")),
         sex = factor(sex, levels = c("Female", "Male")),
         group = factor(group, levels = c("SED", paste0(2 ^ (0:3), "W")))) %>% 
  pivot_longer(cols = c(emmean, estimate),
               names_to = "estimate_type",
               values_to = "estimate",
               values_drop_na = TRUE) %>%
  pivot_longer(cols = contains(".ratio"), 
               names_to = "statistic_type", 
               values_to = "statistic", 
               values_drop_na = TRUE) %>% 
  left_join(model_df, by = "response") %>% 
  mutate(estimate_type = ifelse(estimate_type == "emmean",
                                "(post - pre) mean", 
                                "signed mean (post - pre) rank"),
         statistic_type = sub("\\.ratio", "", statistic_type),
         model_type = ifelse(statistic_type == "W", 
                             "wilcox.test", model_type)) %>% 
  dplyr::select(response, age, sex, group, estimate_type, estimate, SE, 
                lower.CL, upper.CL, statistic_type, statistic, df, n, p.value, 
                p.adj, signif, everything()) %>% 
  arrange(response, age, sex, group)
```

See `?PRE_POST_STATS` for details.

```{r}
print.data.frame(head(PRE_POST_STATS, 10))
```

```{r, eval=FALSE, include=FALSE}
# Save data
usethis::use_data(PRE_POST_STATS, internal = FALSE, overwrite = TRUE,
                  compress = "bzip2", version = 3)
```

# Session Info

```{r}
sessionInfo()
```