An Introduction to t_TOST

In TOSTER: Two One-Sided Tests (TOST) Equivalence Testing

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
library(TOSTER)
library(ggplot2)
library(ggdist)

In an effort to make TOSTER more informative and easier to use, I created the functions t_TOST and simple_htest. These function operates very similarly to base R's t.test function with a few exceptions. First, t_TOST performs 3 t-tests (one two-tailed and two one-tailed tests). Second, simple_htest allows you to run equivalence testing or minimal effects testing using a t-test or Wilcoxon-Mann-Whitney tests using the alternative argument and the output is the same as t.test or wilcox.test (in that the object is of the class htest). In addition, these functions have a generic method where two vectors can be supplied or a formula can be given (e.g.,y ~ group). These functions make it easier to switch between types of t-tests. All three types (two sample, one sample, and paired samples) can be performed/calculated from the same function. Moreover, the summary information and visualizations have been upgraded. This should make the decisions derived from the function more informative and user-friendly.

These functions are not limited to equivalence tests. Minimal effects testing (MET) is possible. MET is useful for situations where the hypothesis is about a minimal effect and the null hypothesis is equivalence.

ggplot() +
  geom_vline(aes(xintercept = -.5),
             linetype = "dashed") +
  geom_vline(aes(xintercept = .5),
             linetype = "dashed") +
  geom_text(aes(
    y = 1,
    x = -0.5,
    vjust = -.9,
    hjust = "middle"
  ),
  angle = 90,
  label = 'Lower Bound') +
  geom_text(aes(
    y = 1,
    x = 0.5,
    vjust = 1.5,
    hjust = "middle"
  ),
  angle = 90,
  label = 'Upper Bound') +
  geom_text(aes(
    y = 1,
    x = 0,
    vjust = 1.5,
    hjust = "middle"
  ),
  #alignment = "center",
  label = "H0"
  ) +
  geom_text(aes(
    y = 1,
    x = 1.5,
    vjust = 1.5,
    hjust = "middle"
  ),
  #alignment = "center",
  label = "H1"
  ) +
  geom_text(aes(
    y = 1,
    x = -1.5,
    vjust = 1.5,
    hjust = "middle"
  ),
  #alignment = "center",
  label = "H1"
  ) +
theme_tidybayes() +
  scale_y_continuous(limits = c(0,1.75)) +
  scale_x_continuous(limits = c(-2,2)) +
  labs(x = "", y = "",
       title="Minimal Effect Test",
       caption = "H1 = Alternative Hypothesis \n H0 = Null Hypothesis") +
  theme(
    strip.text = element_text(face = "bold", size = 10),
    axis.text.y = element_blank(),
    axis.ticks.y = element_blank()
  )

ggplot() +
  geom_vline(aes(xintercept = -.5),
             linetype = "dashed") +
  geom_vline(aes(xintercept = .5),
             linetype = "dashed") +
  geom_text(aes(
    y = 1,
    x = -0.5,
    vjust = -.9,
    hjust = "middle"
  ),
  angle = 90,
  label = 'Lower Bound') +
  geom_text(aes(
    y = 1,
    x = 0.5,
    vjust = 1.5,
    hjust = "middle"
  ),
  angle = 90,
  label = 'Upper Bound') +
  geom_text(aes(
    y = 1,
    x = 0,
    vjust = 1.5,
    hjust = "middle"
  ),
  #alignment = "center",
  label = "H1"
  ) +
  geom_text(aes(
    y = 1,
    x = 1.5,
    vjust = 1.5,
    hjust = "middle"
  ),
  #alignment = "center",
  label = "H0"
  ) +
  geom_text(aes(
    y = 1,
    x = -1.5,
    vjust = 1.5,
    hjust = "middle"
  ),
  #alignment = "center",
  label = "H0"
  ) +
theme_tidybayes() +
  scale_y_continuous(limits = c(0,1.75)) +
  scale_x_continuous(limits = c(-2,2)) +
  labs(x = "",
       y = "",
       title="Equivalence Test",
       caption = "H1 = Alternative Hypothesis \n H0 = Null Hypothesis") +
  theme(
    strip.text = element_text(face = "bold", size = 10),
    axis.text.y = element_blank(),
    axis.ticks.y = element_blank()
  )

In the general introduction to this package, we detailed how to look at old results and how to apply TOST to interpreting those results. However, in many cases, users may have new data that needs to be analyzed. Therefore, t_TOST and simple_htest can be applied to new data. This vignette will use the iris and the sleep data.

data('sleep')
data('iris')

Independent Groups

For this example, we will use the sleep data. In this data there is a group variable and an outcome extra.

head(sleep)

We will assume the data are independent, and that we have equivalence bounds of +/- 0.5 raw units. All we need to do is provide the formula, data, and eqb arguments for the function to run appropriately. In addition, we can set the var.equal argument (to assume equal variance), and the paired argument (sets if the data is paired or not). Both are logical indicators that can be set to TRUE or FALSE. The alpha is automatically set to 0.05 but this can also be adjusted by the user. The Hedges correction is also automatically calculated, but this can be overridden with the bias_correction argument. The hypothesis is automatically set to "EQU" for equivalence but if a minimal effect is of interest then "MET" can be supplied. Note: for this example, we will set smd_ci to "goulet" since it will reduce the time to produce plots.

res1 = t_TOST(formula = extra ~ group,
              data = sleep,
              eqb = .5,
              smd_ci = "goulet")

res1a = t_TOST(x = subset(sleep,group==1)$extra,
               y = subset(sleep,group==2)$extra,
               eqb = .5)

We can also using the "simpler" approach with simple_htest.

# Simple htest

res1b = simple_htest(formula = extra ~ group,
                     data = sleep,
                     mu = .5, # set equivalence bound
                     alternative = "e")

Once the function has run, we can print the results with the print command. This provides a verbose summary of the results.

# t_TOST
print(res1)

# htest

print(res1b)

Plots

Another nice feature is the generic plot method that can provide a visual summary of the results (only available for t_TOST). All of the plots in this package were inspired by the concurve R package. There are two types of plots that can be produced. The first, and default, is the consonance density plot (type = "cd").

plot(res1, type = "cd")

The shading pattern can be modified with the ci_shades.

# Set to shade only the 90% and 95% CI areas
plot(res1, type = "cd",
     ci_shades = c(.9,.95))

Consonance plots, where all confidence intervals can be simultaneous plotted, can also be produced. The advantage here is multiple confidence interval lines can plotted at once.

plot(res1, type = "c",
     ci_lines =  c(.9,.95))

Descriptions

A description of the results can also be produced with the describe or describe_htest method and function respectively.

describe(res1)

describe_htest(res1b)

r describe(res1)

r describe_htest(res1b)

Paired Samples

To perform a paired samples TOST, the process does not change much. We could process the test the same way by providing a formula. All we would need to then is change paired to TRUE.

res2 = t_TOST(formula = extra ~ group,
              data = sleep,
              paired = TRUE,
              eqb = .5)
res2

res2b = simple_htest(
  formula = extra ~ group,
  data = sleep,
  paired = TRUE,
  mu = .5,
  alternative = "e")
res2b

However, we may have two vectors of data that are paired. So we may want to just provide those separately rather than using a data set and setting the formula. This can be demonstrated with the "iris" data.

res3 = t_TOST(x = iris$Sepal.Length,
              y = iris$Sepal.Width,
              paired = TRUE,
              eqb = 1)
res3

res3a = simple_htest(
  x = iris$Sepal.Length,
  y = iris$Sepal.Width,
  paired = TRUE,
  mu = 1,
  alternative = "e"
)
res3a

We may want to perform a Minimal Effect Test with the hypothesis argument set to "MET".

res_met = t_TOST(x = iris$Sepal.Length,
              y = iris$Sepal.Width,
               paired = TRUE,
               hypothesis = "MET",
               eqb = 1,
              smd_ci = "goulet")
res_met

res_metb = simple_htest(x = iris$Sepal.Length,
                       y = iris$Sepal.Width,
                       paired = TRUE,
                       mu = 1,
                       alternative = "minimal.effect")
res_metb

Descriptions

A description of the results can also be produced with the describe or describe_htest method and function respectively.

describe(res_met)

describe_htest(res_metb)

r describe(res_met)

r describe_htest(res_metb)

One Sample t-test

In other cases we may just have a one sample test. If that is the case all we have to do is supply the x argument for the data. For this test we may hypothesis that the mean of Sepal.Length is not more than 5.5 points greater or less than 8.5.

res4 = t_TOST(x = iris$Sepal.Length,
              hypothesis = "EQU",
              eqb = c(5.5,8.5),
              smd_ci = "goulet")
res4

Only have the summary statistics? No problem!

In some cases you may only have access to the summary statistics. Therefore, we created a function, tsum_TOST, to perform the same tests just based on the summary statistics. This involves providing the function with a number of different arguments.

• n1 & n2 the sample sizes (only n1 needs to be provided for one sample case)
• m1 & m2 the sample means
• sd1 & sd2 the sample standard deviation
• r12 the correlation between the paired samples; only needed if paired is set to TRUE

The results from above can be replicated with the tsum_TOST

res_tsum = tsum_TOST(
  m1 = mean(iris$Sepal.Length, na.rm=TRUE),
  sd1 = sd(iris$Sepal.Length, na.rm=TRUE),
  n1 = length(na.omit(iris$Sepal.Length)),
  hypothesis = "EQU",
  eqb = c(5.5,8.5)
)

res_tsum

plot(res_tsum)

describe(res_tsum)

Power Analysis for t-test based TOST

We also created power_t_TOST to allow for power calculations for TOST analyses that utilize t-tests. This function uses a more accurate method than the older functions in TOSTER and match the results of the commercially available PASS software. The exact calculations of power are based on Owen’s Q-function or by direct integration of the bivariate non-central t-distribution^[Inspired by @PowerTOST in the PowerTOST R package. Please see this package for more options]. Approximate power is implemented via the non-central t-distribution or the ‘shifted’ central t-distribution [@phillips1990, @diletti1991]. The function is limited to power analyses involves one sample, two sample, and paired sample cases. More options are available in the PowerTOST R package.

The interface for this function is quite simple and was intended to mimic the base R function power.t.test. The user must specify the 2 equivalence bounds, and leave only one of the other options blank (alpha, power, or n). The "true difference" can be set with delta and the standard deviation (default is 1) can be set with the sd argument. Once everything is set and the function is run, a object of the power.htest class will be returned.

As an example, let's say we are looking at an equivalence study where we assume the true difference is at least 1 unit, the standard deviation is 2.5, and we set the equivalence bounds to 2.5 units as well. If we want to find the sample size adequate to have 95% power at an alpha of 0.025 we enter the following:

power_t_TOST(n = NULL,
  delta = 1,
  sd = 2.5,
  eqb = 2.5,
  alpha = .025,
  power = .95,
  type = "two.sample")

From the analysis above we would conclude that adequate power is achieved with 74 participants per group and 148 participants in total.

References

TOSTER documentation built on Sept. 15, 2023, 1:09 a.m.