---
title: "Agreement Tests"
author: "Aaron R. Caldwell"
date: "Last Updated: `r Sys.Date()`"
output: 
  rmarkdown::html_vignette:
    toc: true
    toc_depth: 2
bibliography: refs.bib
link-citations: true
vignette: >
  %\VignetteIndexEntry{Agreement Tests}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

The `SimplyAgree` R package was created to make the process of quantifying measurement agreement, consistency, and reliability. This package expands upon the capabilities of currently available R packages (such as `psych` and `blandr`) by 1) providing support for agreement studies that involve multiple observations (`agree_test` and `agree_reps` functions) 2) provide robust tests of agreement even in simple studies (`shieh_test` output from the `agree_test` function) and 3) a robust set of reliability statistics (`reli_stats` function).

In this vignette I will briefly demonstrate the implementation of the functions that include tests of agreement. *Please note*, I would recommend using the `agreement_limit` function over these older functions with the exception of `agree_np` which provides non-parametric agreement limits.

```{r}
library(SimplyAgree)
```

# Simple Agreement

## `agree_test`

In the simplest scenario, a study may be conducted to compare one measure (e.g., `x`) and another (e.g., `y`). In this scenario each pair of observations (x and y) are *independent*; meaning that each pair represents one subject/participant. In most cases we have a degree of agreement that we would deem adequate. This may constitute a hypothesis wherein you may believe the agreement between two measurements is within a certain limit (limits of agreement). If this is the goal then the `agree_test` function is what you need to use in this package.

The data for the two measurements are put into the `x` and `y` arguments. If there is a hypothesized limit of agreement then this can be set with the `delta` argument (this is optional). Next, the limit of agreement can be set with the `agree.level` and the confidence level ($1-\alpha$). Once those are set the analysis can be run. Please note, this package has pre-loaded data from the Zou 2013 paper. While data does not conform the assumptions of the test it can be used to test out many of the functions in this package. Since there isn't an *a priori* hypothesis I will not declare a `delta` argument, but I will estimate the 95% confidence intervals for 80% limits of agreement.

```{r}
a1 = agree_test(x = reps$x,
                y = reps$y,
                agree.level = .8)
```

We can then print the general results. These results include the general parameters of the analysis up top, then the results of the Shieh exact test for agreement (no conclusion is included due to the lack of a `delta` argument being set). Then the limits of agreement, with confidence limits, are included. Lastly, Lin's Concordance Correlation Coefficient, another measure of agreement, is also included.

```{r}
print(a1)
```

Next, we can use the generic `plot` function to produce visualizations of agreement. This includes the Bland-Altman plot (`type = 1`) and a line-of-identity plot (`type = 2`).

```{r, fig.width=6, fig.height=6}
plot(a1, type = 1)

plot(a1, type = 2) 
```


## Calculations in for Simple Agreement

### Shieh's test

The hypothesis test procedure is based on the "exact" approach details by @shieh2019. In this procedure the null hypothesis (not acceptable agreement) is rejected if the extreme lower bound and upper bound are within the proposed agreement limits. The agreement limits, $\hat\theta_{EL}$ and $\hat\theta_{EU}$ respectively, are calculated as the following:

$$
\hat\theta_{EL,EU} = \bar{d} \pm \gamma_{1-\alpha}\cdot \frac{S}{\sqrt{N}}
$$

wherein $\bar{d}$ is the mean difference between the two methods, $S$ is the standard deviation of the differences, $N$ is the total number of pairs, and $\gamma_{1-\alpha}$ critical value (which requires a specialized function within `R` to estimate).

### Limits of Agreement

The reported limits of agreement are derived from the work of @bland1986 and @bland1999.

**LoA**

$$
LoA = \bar d \pm z_{1-(1-agree)/2} \cdot S_d
$$
wherein $z_{1-(1-agree)/2}$ is the value of the normal distribution at the given agreement level (default is 95%), $\bar d$ is the mean of the differences, and $S_d$ is the standard deviations of the differences.

**Confidence Interval**

$$
LoA_{C.I.} = LoA \pm t_{1-\alpha/2,N-1} \cdot \sqrt{\left[\frac{1}{N}+\frac{(z_{1-\alpha/2})^{2}}{2 \cdot (N-1)} \right] \cdot S^2}
$$

wherein, $t$ is the critical t-value at the given sample size and confidence level (`conf.level`), $z$ is the value of the normal distribution at the given alpha level (`agree.level`), and $S^2$ is the variance of the difference scores. If `TOST` is set to TRUE then equation is altered slightly with the critical t ($t_{1-\alpha,N-1}$).

### Concordance Correlation Coefficient

The CCC was calculated as outlined by @lin1989 (with later corrections).

$$
\hat\rho_c = \frac{2 \cdot s_{xy}} {s_x^2 + s_y^2+(\bar x-\bar y)^2}
$$
where $s_{xy}$ is the covariance, $s_x^2$ and $s_y^2$ are the variances of x and y respectively, and $(\bar x-\bar y)$ is the difference in the means of x & y.

# Repeated Measures Agreement

In many cases there are multiple measurements taken within subjects when comparing two measurements tools. In some cases the true underlying value will not be expected to vary (i.e., replicates; `agree_reps`), or multiple measurements may be taken within an individual *and* these values are expected to vary (i.e., nested design; `agree_nest`).

The confidence limits on the limits of agreement are based on the "MOVER" method described in detail by @zou2011. However, both functions operate similarly to `agree_test`; the only difference being that the data has to be provided as a `data.frame` in R.

## `agree_reps`

This function is for cases where the underlying values do *not* vary within subjects. This can be considered cases where replicate measure may be taken. For example, a researcher may want to compare the performance of two ELISA assays where measurements are taken in duplicate/triplicate.

So, for this function you will have to provide the data frame object with the `data` argument and the names of the columns containing the first (`x` argument) and second (`y` argument) must then be provided. An additional column indicating the subject identifier (`id`) must also be provided. Again, if there is a hypothesized agreement limit then this could be provided with the `delta` argument.

```{r}
a2 = agree_reps(x = "x",
                y = "y",
                id = "id",
                data = reps,
                agree.level = .8)
```

The results can then be printed. The printing format is very similar to `agree_test`, but notice that 1) the hypothesis test is based on the limits of agreement (MOVER method), 2) the Concordance Correlation Coefficient is calculated via the U-statistics method, 3) the Shieh TOST results are missing because they cannot be estimated for this type of design.

```{r}
print(a2)
```
```{r, fig.width=6, fig.height=6}
plot(a2, type = 1)

plot(a2, type = 2)
```

## `agree_nest`

This function is for cases where the underlying values may vary within subjects. This can be considered cases where there are distinct pairs of data wherein data is collected in different times/conditions within each subject. An example would be measuring blood pressure on two different devices on many people at different time points/days.

The function works almost identically to `agree_reps` but the underlying calculations are different


```{r}
a3 = agree_nest(x = "x",
                y = "y",
                id = "id",
                data = reps,
                agree.level = .8)
```

The printed results (and plots) are very similar to `agree_reps`. However, the CCC result now has a warning because the calculation in this scenario may not be entirely appropriate given the nature of the data.

```{r}
print(a3)
```
```{r, fig.width=6, fig.height=6}
plot(a3, type = 1)

plot(a3, type = 2)
```

## Calculations for `agree_reps` & `agree_nest`

All the calculations for the limits of agreement in these two functions can be found in the article by @zou2011.

## `agree_nest` LoA

**Step 1: Calculate Individual Subject Means and Variances**

$$
\bar x_i = \frac{1}{n_{xi}} \Sigma_{j=1}^{n_{xi}} x_{ij}
$$
$$
\bar y_i = \frac{1}{n_{yi}} \Sigma_{j=1}^{n_{yi}} y_{ij}
$$
$$
\bar d_i = \bar x_i - \bar y_i
$$

$$
\bar d = \Sigma_{i=1}^{n}\frac{d_i}{n} 
$$

$$
s_{xi}^2 = \Sigma_{j=1}^{n_{xi}} \frac{(x_{xj}- \bar x_i)^2}{n_{xi}-1}
$$

$$
s_{yi}^2 = \Sigma_{j=1}^{n_{yi}} \frac{(y_{ij}- \bar y_i)^2}{n_{yi}-1}
$$

$$
s_{\bar d}^2 = \Sigma_{j=1}^{n} \frac{(d_{i}- \bar d)^2}{n-1}
$$

**Step 2: Compute pooled estimates of within subject errors**

$$
s^2_{xw} = \Sigma_{i=1}^{n} [\frac{n_{xi} -1}{N_x -1} \cdot s^2_{xi}]
$$

$$
s^2_{yw} = \Sigma_{i=1}^{n} [\frac{n_{yi} -1}{N_y -1} \cdot s^2_{yi}]
$$

**Step 3: Compute Harmonic Means of Replicates**

$$
m_{xh} = \frac{n}{\Sigma_{i=1}^n \frac{1}{n_{xi}}}
$$

$$
m_{yh} = \frac{n}{\Sigma_{i=1}^n \frac{1}{n_{yi}}}
$$
**Step 4: Compute the variance of the differences**

$$
s^2_d = s^2_{\bar d} + (1+\frac{1}{m_{xh}}) \cdot s^2_{xw} + (1+\frac{1}{m_{yh}}) \cdot s^2_{yw}
$$
**Step 5: Compute MOVER Components**

$$
S_{11} = s_{\bar d}^2 \cdot (1 - \frac{n-1}{\chi^2_{(1-\alpha, n-1)}})
$$

$$
S_{12} = (1-\frac{1}{m_{xh}}) \cdot (1 - \frac{N_x-n}{\chi^2_{(1-\alpha, N_x-n)}}) \cdot s^2_{xw}
$$
$$
S_{13} = (1-\frac{1}{m_{yh}}) \cdot (1 - \frac{N_y-n}{\chi^2_{(1-\alpha, N_y-n)}}) \cdot s^2_{yw}
$$

$$
S_1 = \sqrt{S_{11}^2 +S_{12}^2 +S_{13}^2}
$$
$$
l = s_d^2 - S_1
$$

$$
u = s_d^2 + S_1
$$

$$
LME = \sqrt{\frac{z^2_{\alpha} \cdot s_d^2}{n} + z^2_{\beta/2} \cdot(\sqrt{u} - \sqrt{s^2_d})^2}
$$

$$
RME = \sqrt{\frac{z^2_{\alpha} \cdot s_d^2}{n} + z^2_{\beta/2} \cdot(\sqrt{l} - \sqrt{s^2_d})^2}
$$
### LoA

$$
LoA_{lower} = \bar d - z_{\beta/2} \cdot s_d
$$

$$
LoA_{upper} = \bar d + z_{\beta/2} \cdot s_d
$$

### Lower LoA CI

$$
Lower \space CI = LoA_{lower} - LME
$$


$$
Upper \space CI = LoA_{lower} + RME
$$

### Upper LoA CI

$$
Lower \space CI = LoA_{upper} - RME
$$


$$
Upper \space CI = LoA_{upper} + LME
$$

## `agree_reps` LoA

### LoA

**Step 1: Compute mean and variance**

$$
\bar d_i = \Sigma_{j=1}^{n_i} \frac{d_{ij}}{n_i}
$$
$$
\bar d = \Sigma^{n}_{i=1} \frac{d_i}{n}
$$

$$
s_i^2 = \Sigma_{j=1}^{n_i} \frac{(d_{ij} - \bar d_i)^2}{n_i-1}
$$
**Step 2: Compute pooled estimate of within subject error**

$$
s_{dw}^2 = \Sigma_{i=1}^{n} [\frac{n_i-1}{N-n} \cdot s_i^2]
$$

**Step 3: Compute pooled estimate of between subject error**

$$
s^2_b = \Sigma_{i=1}^n \frac{ (\bar d_i - \bar d)^2}{n-1}
$$

**Step 4: Compute the harmonic mean of the replicate size**

$$
m_h = \frac{n}{\Sigma_{i=1}^n m_i^{-1}}
$$

**Step 5: Compute SD of the difference **

$$
s_d^2 = s^2_b + (1+m_h^{-1}) \cdot s_{dw}^2
$$
**Step 6: Calculate l and u**

$$
l = s_d^2 - \sqrt{[s_d^2 \cdot (1 - \frac{n-1}{\chi^2_{(1-\alpha, n-1)}})]^2+[(1-m_h^{-1}) \cdot (1- \frac{N-n}{\chi^2_{(1-\alpha, N-n)}})]^2}
$$
$$
u = s_d^2 + \sqrt{[s_d^2 \cdot (1 - \frac{n-1}{\chi^2_{(1-\alpha, n-1)}})]^2+[(1-m_h^{-1}) \cdot (1- \frac{N-n}{\chi^2_{(1-\alpha, N-n)}})]^2}
$$
**Step 7: Compute LME and RME**

$$
LME = \sqrt{\frac{z_{\alpha} \cdot s_d^2}{n} + z_{\beta/2}^2 \cdot (\sqrt{u}-\sqrt{s^2_d} )^2}
$$

$$
RME = \sqrt{\frac{z_{\alpha} \cdot s_d^2}{n} + z_{\beta/2}^2 \cdot (\sqrt{l}-\sqrt{s^2_d} )^2}
$$


### LoA

$$
LoA_{lower} = \bar d - z_{\beta/2} \cdot s_d
$$


$$
LoA_{upper} = \bar d + z_{\beta/2} \cdot s_d
$$

### Lower LoA CI

$$
Lower \space CI = LoA_{lower} - LME
$$


$$
Upper \space CI = LoA_{lower} + RME
$$

### Upper LoA CI

$$
Lower \space CI = LoA_{upper} - RME
$$


$$
Upper \space CI = LoA_{upper} + LME
$$

## CCC from U-statistics

The CCC calculations are derived from the `cccUst` function of the `cccrm` R package. The mathematics for this CCC calculation can be found in the work of @king2007 and @carrasco2009.

# Checking Assumptions

The assumptions of normality, heteroscedasticity, and proportional bias can all be checked using the `check` method.

The function will provide 3 plots: Q-Q normality plot, standardized residuals plot, and residuals plot.

All 3 plots will have a statistical test in the bottom right corner. The Shapiro-Wilk test is included for the normality plot, the Bagan-Preusch test for heterogeneity, and the test for linear slope on the residuals plot.

## An Example

```{r}
a1 = agree_test(x = reps$x,
                y = reps$y,
                agree.level = .8)

check(a1)
```


# Proportional Bias

As the check plots for `a1` show, proportional bias can sometimes occur. In these cases @bland1999 recommended adjusting the bias and LoA for the proportional bias. This is simply done by include a slope for the average of both measurements (i.e, using an intercept + slope model rather than intercept only model).

For any of the "agree" functions, this can be accomplished with the `prop_bias` argument. When this is set to TRUE, then the proportional bias adjusted model is utilized. However, you should be careful with interpreting the hypothesis tests in these cases because the results are likely bogus for the extreme ends of the measurement. In any case, plots of the data should always be inspected

```{r}
a1 = agree_test(x = reps$x,
                y = reps$y,
                prop_bias = TRUE,
                agree.level = .8)
print(a1)

plot(a1)
```


# Non-parametric Alternatives

When the assumptions of any of the tests above are violated then a non-parametric alternative may be useful. The `agree_np` function is the non-parametric alternative for the `SimpleAgree` R package, and is largely based on the recommendations of Bland & Altman [@bland1999, pg. 157].

The function performs two tests:

1. A binomial test on whether or not the observed differences are within the maximal allowable differences bounds (`delta` argument).
2. Quantile regression, derived from the `quantreg` package [@quantreg], two estimate the median and 95% limits of agreement. If the default agreement levels are used (`agree.level = .95`) then the 0.025, 0.5 (median), and 0.975 quantiles are estimated.

The function also *requires* the `delta` argument (otherwise the binomial test would be useless). Otherwise, it functions just like the other agreement functions.

In the code demo below, you will notice that the limits of agreement are no longer symmetric around the bias estimate.

```{r}
a1 = agree_np(x = "x",
              y = "y",
              data = reps,
              delta = 2,
              prop_bias = FALSE,
              agree.level = .8)
print(a1)

plot(a1)
```


We can also perform the same analysis with proportional bias no longer assumed. You can see that the LoA changes but the test for agreement does not.

```{r}
a1 = agree_np(x = "x",
              y = "y",
              data = reps,
              delta = 2,
              prop_bias = TRUE,
              agree.level = .8)
print(a1)

plot(a1)

```


# "Big" Data

Sometimes there may be a lot of data and individual points of data on Bland-Altman plot may be less than ideal. In order to change the plots from showing the individual data points we can modify the `geom_point` argument.

```{r}
set.seed(81346)
x = rnorm(750, 100, 10)
diff = rnorm(750, 0, 1)
y = x + diff

df = data.frame(x = x,
                y = y)

a1 = agree_test(x = df$x,
                y = df$y,
                #prop_bias =TRUE,
                agree.level = .95)

plot(a1, 
     geom = "geom_point")

plot(a1, 
     geom = "geom_bin2d")

plot(a1,
     geom = "geom_density_2d")

plot(a1,
     geom = "geom_density_2d_filled")

plot(a1,
     geom = "stat_density_2d")
```


# Mixed Models

In some cases, the agreement calculations involve comparing two methods within individuals within varying conditions. For example, the "recpre_long" data set within this package contains two measurements of rectal temperature in 3 different conditions (where there is a fixed effect of condition). For this particular case we can use bootstrapping to estimate the limits of agreement.

The `loa_lme` function can then calculate the limits of agreement. Like the previous functions, the data set must be set with the `data` argument. The `diff` is the column which contains the difference between the two measurements. The `condition` is the column that indicates the different conditions that the measurements were taken within. The `id` is the column containing the subject/participant identifier. The final two arguments `replicates` and `type` set the requirements for the bootstrapping procedure. **Warning**: This is a computationally heavy procedure and it may take a few seconds to a few minutes to complete depending the number of replicates or if `het_var` is set to TRUE.

```{r,warning=FALSE,eval=FALSE}

recpre_long$avg = (recpre_long$PM + recpre_long$PM)/2
a4 = loa_lme(data = recpre_long,
               diff = "diff",
               avg = "avg",
               #condition = "trial_condition",
               id = "id",
               #plot.xaxis = "AM",
               replicates = 199,
               type = "perc")
```



# References
