Longitudinal Growth Modeling Options

growthSS(..., form, ...)

The form argument of growthSS needs to specify the outcome variable, the time variable, an identifier for individuals, and the grouping structure. These are passed as a formula object, using similar syntax to lme4 and brms, such as outcome ~ time|individual_id/group_id. Verbally this would be read as “outcome modeled by time accounting for correlation between individual_id’s with fixed effects specified per each group_id”. Note that this formula will not have to change for different growth models, this is only to specify the structure of your data. This simplification requires each of these parts of the formula must be a single column in your dataframe. Note that for each group in your data a set of parameters will be estimated. If you have lots of groups in your data then it may make sense to fit models to only a few groups at a time. If you do that then the models can still be compared by extracting the MCMC draws, combining them into a dataframe, then using brms::hypothesis per normal.

For model backends that do not account for autocorrelation the individual will not be used and can be omitted leaving outcome ~ time|group_id, but there is no harm in including the individual_id component.

growthSS(..., sigma, ...)

The sigma argument controls distributional sub models. This is only used with nlme and brms backends which support different options.

In nlme models this models sigma and can be “int”, “power”, or “exp” which correspond to using nlme::varIdent (constant variance within groups), nlme::varPower (variance changing by a power function), nlme::varExp (variance changing by an exponential function).

In brms models this can be specified in the same way as the general growth model and can be used to model any distributional parameter in the model family you use. The default model family is Student T, which has sigma and nu parameters. Other distributions can be specified in the main growth model as model = "family_name: model_name" (model = "poisson: linear" as an example). For details on available families see ?brms::brmsfamily. There is also an “int” model type which fits a 0 slope intercept only model. While “int” can be used in any brms model the option is meant to be used specifying a homoskedastic sub model, or a period of noise before the main growth trend begins (in terms of growth or variance). Distributional parameters that are not specified will be modeled as constant between groups.

At a high level we can think about any of these models as fitting a curve to these lines.

set.seed(345)
gomp <- growthSim("gompertz", n = 20, t = 35, params = list(
  "A" = c(200, 180, 160),
  "B" = c(20, 22, 18),
  "C" = c(0.15, 0.2, 0.1)
))


sigma_df <- aggregate(y ~ group + time, data = gomp, FUN = sd)

ggplot(sigma_df, aes(x = time, y = y, group = group)) +
  geom_line(color = "gray60") +
  pcv_theme() +
  labs(y = "SD of y", title = "Gompertz Sigma")

Several options are shown here, ignoring grouping here since the data is already aggregated.

draw_gomp_sigma <- function(x) {
  23 * exp(-21 * exp(-0.22 * x))
}
ggplot(sigma_df, aes(x = time, y = y)) +
  geom_line(aes(group = group), color = "gray60") +
  geom_hline(aes(yintercept = 12, color = "Homoskedastic"),
    linetype = 5,
    key_glyph = draw_key_path
  ) +
  geom_abline(aes(slope = 0.8, intercept = 0, color = "Linear"),
    linetype = 5,
    key_glyph = draw_key_path
  ) +
  geom_smooth(
    method = "gam", aes(color = "Spline"), linetype = 5, se = FALSE,
    key_glyph = draw_key_path
  ) +
  geom_function(fun = draw_gomp_sigma, aes(color = "Gompertz"), linetype = 5) +
  scale_color_viridis_d(option = "plasma", begin = 0.1, end = 0.9) +
  guides(color = guide_legend(override.aes = list(linewidth = 1, linetype = 1))) +
  pcv_theme() +
  theme(legend.position = "bottom") +
  labs(y = "SD of y", title = "Gompertz Sigma", color = "")

“int” will specify a homoskedastic model, that is one with constant variance over time per each group. While this is the default for almost every kind of statistical modeling it is an unrealistic assumption in this setting where we often follow growth from small seedlings to potentially fully grown plants. Even if we start with larger plants the homoskedastic assumption almost never holds in longitudinal modeling. We can fit an example model and see the issue with the homoskedastic assumption through the model’s credible intervals, which are far too wide at the beginning of the experiment and even include some negative values for plant area.

ss <- growthSS(
  model = "gompertz", form = y ~ time | id / group, sigma = "int",
  df = gomp, start = list("A" = 130, "B" = 15, "C" = 0.25)
)

fit_h <- fitGrowth(ss, iter = 1000, cores = 4, chains = 4, silent = 0)

brmPlot(fit_h, form = ss$pcvrForm, df = ss$df)

We can relax this assumption and model sigma separately from the main growth trend. To show an example of the options in pcvr, here we repeat the example from above using a linear submodel. Note that here we add some extra controls to the model fitting algorithm to help the model fit well with the added complexity at the cost of being slower.

ss <- growthSS(
  model = "gompertz", form = y ~ time | id / group, sigma = "linear",
  df = gomp, start = list("A" = 130, "B" = 15, "C" = 0.25)
)

fit_l <- fitGrowth(ss,
  iter = 1000, cores = 4, chains = 4, silent = 0,
  control = list(adapt_delta = 0.999, max_treedepth = 20)
)

p1 <- brmPlot(fit_l, form = ss$pcvrForm, df = ss$df)
p2 <- p1 + coord_cartesian(ylim = c(0, 300))
p <- p1 / p2
p

This model is also a poor fit, but it has a different problem. It accurately models the low variability at the beginning of the experiment, but the linear model is not flexible enough to adapt to the changes in variance even in this simulated data.

We can also use spline sub models. The spline model does a very good job of fitting the data due to the natural flexibility of polynomial functions. Again this added accuracy comes at the cost of taking longer for the model to fit. Here we can specify “gam” or “spline” for backwards compatibility.

ss <- growthSS(
  model = "gompertz", form = y ~ time | id / group, sigma = "spline",
  df = gomp, start = list("A" = 130, "B" = 15, "C" = 0.25)
)

fit_s <- fitGrowth(ss,
  iter = 2000, cores = 4, chains = 4, silent = 0,
  control = list(adapt_delta = 0.999, max_treedepth = 20)
)

brmPlot(fit_s, form = ss$pcvrForm, df = ss$df)

Here we try applying a gompertz function to the variance submodel. While this is much less flexible than splines it tends to describe the variance of a sigmoid growth model quite well and allows for easier hypothesis testing between groups. A fringe benefit can also be the predictability of the gompertz formula in extrapolating future data. Splines can have unexpected behavior when trying to predict timepoints outside of your initial data, but the gompertz formula is more predictable. Additionally, since the spline sub model will fit many basis functions this will generally be significantly faster since it only needs to find 3 parameters to complete the sub model, and each can have a mildly informative prior. As a single reference point, the model below fit in about 6 minutes while the spline model above took slightly over an hour to fit. These example models have three groups and the model with a gompertz sub model contains 21 total parameters while the spline sub model version contains 43 total parameters.

When setting priors for the gompertz sub-model it is generally reasonable to expect a similar growth rate and inflection point as in the main model (assuming the main model is gompertz as well).

ss <- growthSS(
  model = "gompertz", form = y ~ time | id / group, sigma = "gompertz",
  df = gomp, start = list(
    "A" = 130, "B" = 15, "C" = 0.25,
    "sigmaA" = 15, "sigmaB" = 15, "sigmaC" = 0.25
  ),
  type = "brms"
)

fit_g <- fitGrowth(ss,
  iter = 2000, cores = 4, chains = 4, silent = 0,
  control = list(adapt_delta = 0.999, max_treedepth = 20)
)

brmPlot(fit_g, form = ss$pcvrForm, df = ss$df)

A few other options are shown here as further examples. There are as many ways to model variance as there are to model growth using the brms backend, but other options are more limited.

draw_gomp_sigma <- function(x) {
  23 * exp(-21 * exp(-0.22 * x))
}
draw_logistic_sigma <- function(x) {
  20 / (1 + exp((15 - x) / 2))
}
draw_logistic_exp <- function(x) {
  2.5 * exp(0.08 * x)
}
draw_logistic_quad <- function(x) {
  (0.3 * x) + (0.02 * x^2)
}

ggplot(sigma_df, aes(x = time, y = y)) +
  geom_line(aes(group = group), color = "gray60", linetype = 5) +
  geom_hline(aes(yintercept = 12, color = "Homoskedastic"), linetype = 1) +
  geom_abline(aes(slope = 0.8, intercept = 0, color = "Linear"),
    linetype = 1,
    key_glyph = draw_key_path
  ) +
  geom_smooth(
    method = "gam", aes(color = "Spline"), linetype = 1, se = FALSE,
    key_glyph = draw_key_path
  ) +
  geom_function(fun = draw_gomp_sigma, aes(color = "Gompertz"), linetype = 1) +
  geom_function(fun = draw_logistic_sigma, aes(color = "Logistic"), linetype = 1) +
  geom_function(fun = draw_logistic_exp, aes(color = "Exponential"), linetype = 1) +
  geom_function(fun = draw_logistic_quad, aes(color = "Quadratic"), linetype = 1) +
  scale_color_viridis_d(option = "plasma", begin = 0.1, end = 0.9) +
  guides(color = guide_legend(override.aes = list(linewidth = 1, linetype = 1))) +
  pcv_theme() +
  theme(legend.position = "bottom") +
  labs(y = "SD of y", title = "Gompertz Sigma", color = "")

When considering several sub models (or growth models) we can compare brms models using Leave-One-Out Information Criterion (LOO IC). For frequentist models a more familiar metric like BIC or AIC might be used.

loo_spline <- add_criterion(fit_s, "loo")
loo_homo <- add_criterion(fit_h, "loo")
loo_linear <- add_criterion(fit_l, "loo")
loo_gomp <- add_criterion(fit_g, "loo")

h <- loo_homo$criteria$loo$estimates[3, 1]
s <- loo_spline$criteria$loo$estimates[3, 1]
l <- loo_linear$criteria$loo$estimates[3, 1]
g <- loo_gomp$criteria$loo$estimates[3, 1]

loodf <- data.frame(loo = c(h, s, l, g), model = c("Homosked", "Spline", "Linear", "Gompertz"))
loodf$model <- factor(loodf$model, levels = unique(loodf$model[order(loodf$loo)]), ordered = TRUE)

ggplot(
  loodf,
  aes(x = model, y = loo, fill = model)
) +
  geom_col() +
  scale_fill_viridis_d() +
  labs(y = "LOO Information Criteria", x = "Sub Model of Sigma") +
  theme_minimal() +
  theme(legend.position = "none")

The spline sub-model tends to have the best LOO IC, but comparing credible intervals while taking speed and interpretability into account may change which model is the best option for your situation. For this particular data the gompertz submodel does seem to perform very well despite the LOO IC difference from using splines.

growthSS(..., start, ...)

One of the main difficulties in non-linear modeling is getting the models to fit without convergence errors. Using growthSS the six main model options (and GAMs, although in a different way) are self-starting and do not require starting values. For the double sigmoid options starting values are required though.

Additionally, using the brms backend this argument is used to specify prior distributions. Setting appropriate prior distributions is an important part and often criticized part of Bayesian statistics. Prior distributions are often talked about in the language of “prior beliefs”, which can be somewhat misleading. Instead it can be helpful to think of prior distributions as hard-headed prior evidence.

In a broad sense, priors can be “strong” or “weak”.

set.seed(345)
ln <- growthSim("linear", n = 5, t = 10, params = list("A" = c(2, 3, 10)))

strongPrior <- prior(student_t(3, 0, 5), dpar = "sigma", class = "b") +
  prior(gamma(2, 0.1), class = "nu", lb = 0.001) +
  prior(normal(10, .05), nlpar = "A", lb = 0)

ss <- growthSS(
  model = "linear", form = y ~ time | id / group, sigma = "homo",
  df = ln, priors = strongPrior
)

fit <- fitGrowth(ss, iter = 1000, cores = 2, chains = 2, silent = 0)

brmPlot(fit, form = ss$pcvrForm, df = ss$df) +
  coord_cartesian(ylim = c(0, 100))

weakPrior <- prior(student_t(3, 0, 5), dpar = "sigma", class = "b") +
  prior(gamma(2, 0.1), class = "nu", lb = 0.001) +
  prior(lognormal(log(10), 0.25), nlpar = "A", lb = 0)

ss <- growthSS(
  model = "linear", form = y ~ time | id / group, sigma = "homo",
  df = ln, priors = weakPrior
)

fit <- fitGrowth(ss, iter = 1000, cores = 2, chains = 2, silent = 0)

brmPlot(fit, form = ss$pcvrForm, df = ss$df) +
  coord_cartesian(ylim = c(0, 100))

priors <- list("A" = 130, "B" = 10, "C" = 0.2)
priorPlots <- plotPrior(priors)
priorPlots[[1]] / priorPlots[[2]] / priorPlots[[3]]

twoPriors <- list("A" = c(100, 130), "B" = c(6, 12), "C" = c(0.5, 0.25))
plotPrior(twoPriors, "gompertz", n = 100)[[1]]

growthSS(..., tau, ...)

For nlrq models the “tau” argument determines which quantiles are fit. By default this uses the median (0.5), but can be any quantile or vector of quantiles between 0 and 1. In the next section we fit an nlrq model with many quantiles shown.

growthSS(..., hierarchy, ...)

Hierarchical models can be specified by adding covariates to the model formula and specifying models for those covariates in the hierarchy argument.

A hierarchical formula would be written as y ~ time + covar | id / group and we could specify to only model the A parameter as being modeled by covar by adding hierarchy = list("A" = "int_linear").

This would change a logistic model from something like:

\[ Y \sim \frac{A}{(1 + \text{e}^{( (B-x)/C)})}\\ A \sim 0 + \text{group}\\ B \sim 0 + \text{group}\\ C \sim 0 + \text{group} \]

to being something like:

\[ Y \sim \frac{A}{(1 + \text{e}^{( (B-x)/C)})}\\ A \sim AI + AA \cdot \text{covariate}\\ B \sim 0 + \text{group}\\ C \sim 0 + \text{group}\\ AI \sim 0 + \text{group}\\ AA \sim 0 + \text{group} \]

This can be helpful in modeling the effect of time on one phenotype given another or in including watering data, etc.

Check Model Fit

We can check the model fits using growthPlot.

growthPlot(nls_fit, form = nls_ss$pcvrForm, df = nls_ss$df)

growthPlot(nlrq_fit, form = nlrq_ss$pcvrForm, df = nlrq_ss$df)

growthPlot(nlme_fit, form = nlme_ss$pcvrForm, df = nlme_ss$df)

growthPlot(mgcv_fit, form = mgcv_ss$pcvrForm, df = mgcv_ss$df)

growthPlot(brms_fit, form = brms_ss$pcvrForm, df = brms_ss$df)

testGrowth

testGrowth takes three arguments, the growthSS output used to fit a model, the model itself, and a list of parameters to test. Broadly, two kinds of tests are supported. First, if the test argument is a parameter name or a vector of parameter names then a version of the model is fit with the parameters in test not varying by group and the models are compared with an anova. In this case the resulting p-value is broadly testing the null hypothesis that the groups have the same value for that parameter.

Here we see that for our nls model is statistically significantly improved by varying asymptote by group.

testGrowth(nls_ss, nls_fit, test = "A")$anova

## Analysis of Variance Table
## 
## Model 1: y ~ A/(1 + exp((B[group] - time)/C[group]))
## Model 2: y ~ A[group]/(1 + exp((B[group] - time)/C[group]))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1    995     204775                                
## 2    994     171686  1  33089  191.57 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Likewise for the 49th percentile in our nlrq model

testGrowth(nlrq_ss, nlrq_fit, test = "A")[["0.49"]]

## Model 1: y ~ A[group]/(1 + exp((B[group] - time)/C[group]))
## Model 2: y ~ A/(1 + exp((B[group] - time)/C[group]))
##   #Df  LogLik Df  Chisq Pr(>Chisq)    
## 1   6 -3901.2                         
## 2   5 -4029.8 -1 257.27  < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

And the same is shown in our nlme model.

testGrowth(nlme_ss, nlme_fit, test = "A")$anova

##         Model df      AIC      BIC    logLik   Test  L.Ratio p-value
## nullMod     1 13 4923.097 4986.898 -2448.548                        
## fit         2 16 4913.518 4992.042 -2440.759 1 vs 2 15.57932  0.0014

We cannot test parameters in the GAM of course but we still see that the grouping improves the model fit.

testGrowth(mgcv_ss, mgcv_fit)$anova

## Analysis of Deviance Table
## 
## Model 1: y ~ s(time)
## Model 2: y ~ 0 + group + s(time, by = group)
##   Resid. Df Resid. Dev     Df Deviance     F    Pr(>F)    
## 1    992.43     271044                                    
## 2    985.06     172134 7.3716    98909 76.96 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

These are relatively basic options for testing hypotheses in non-linear growth models and should serve as a good starting point for frequentist model analysis.

The second kind of hypothesis test in testGrowth is used if the test argument is a hypothesis or list of hypotheses similar to those used in brms::hypothesis syntax. These can test more complex non-linear hypotheses by using car::deltaMethod to calculate standard errors for model parameters. Note that this kind of testing is only supported with nls and nlme model backends. This is a much more flexible testing option.

Here we test several hypotheses on our nls and nlme models to show the versatility of this option. In practice you should not use these hypotheses and should be giving some thought to how to express your stated hypothesis in terms of the model parameters.

testGrowth(fit = nls_fit, test = list(
  "A1 - A2 *1.1",
  "(B1+1) - B2",
  "C1 - (C2-0.5)",
  "A1/B1 - (1.1 * A2/B2)"
))

##                    Form   Estimate        SE   t-value      p-value
## 1          A1 - A2 *1.1 19.9337491 2.5232917  7.899899 7.368000e-15
## 2           (B1+1) - B2  3.1348883 0.1624348 19.299359 9.834919e-71
## 3         C1 - (C2-0.5)  0.1517508 0.1292836  1.173782 2.407636e-01
## 4 A1/B1 - (1.1 * A2/B2) -1.2329672 0.1538017  8.016603 3.036365e-15

testGrowth(fit = nlme_fit, test = list(
  "(A.groupa / A.groupb) - 0.9",
  "1 + (B.groupa - B.groupb)",
  "C.groupa/C.groupb - 1"
))

##                          Form   Estimate         SE   Z-value      p-value
## 1 (A.groupa / A.groupb) - 0.9  0.3122200 0.03541821  8.815237 1.194318e-18
## 2   1 + (B.groupa - B.groupb)  3.1217633 0.16951043 18.416350 9.713729e-76
## 3       C.groupa/C.groupb - 1 -0.1002475 0.03665857  2.734627 6.245101e-03

brms::hypothesis

With the brms backend our options expand dramatically. We can write arbitrarily complex hypotheses about our models and test the posterior probability using brms::hypothesis. Here we keep it simple and test that group A has an asymptote 10% larger than group B’s asymptote. If we had a more complicated dataset then there is no reason we could not expand this hypothesis instead to something like ((A_genotype1_treatment1/A_genotype1_treatment2))-(1.1*(A_genotype2_treatment1/A_genotype2_treatment2)) > 0 to test relative tolerance to different treatments between genotypes. It would be out of scope to go over all the potentially interesting hypotheses since those will depend on your experimental design and questions, but hopefully this communicates the flexibility in using these models once they are fit. Access to the brms::hypothesis function is generally a compelling reason to use the brms backend if you have questions beyond “are these groups different?”

(hyp <- brms::hypothesis(brms_fit, "(A_groupa) > 1.1 * (A_groupb)"))

##                          Hypothesis Estimate Est.Error CI.Lower CI.Upper
## 1 ((A_groupa))-(1.1*(A_groupb)) > 0 16.78802  3.135185 12.01271  21.9525
##   Evid.Ratio Post.Prob Star
## 1        Inf         1    *