---
title: "Technical Error of Measurement"
author: "Ernest Guevarra"
date: "18 December 2024"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Technical Error of Measurement}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

## Intra-observer TEM for two measurements, and inter-observer TEM involving two measurers

$$ TEM ~ = ~ \sqrt{\frac{\sum D ^ 2}{2N}} $$
where

$D ~ = ~ \text{The difference between measurements}$

$N ~ = ~ \text{The number of individuals measured}$


## Inter-observer TEM with more than two observers

$$ TEM ~ = ~ \sqrt{\sum\nolimits_1 ^ N \left (\sum\nolimits_1 ^ K  M ^ 2 ~ - ~ \frac{\left (\sum\nolimits_1 ^ K M \right ) ^ 2 / K}{N(K ~ - ~ 1)} \right )}   $$
where

$N ~ = ~ \text{Number of subjects}$

$K ~ = ~ \text{Number of observers}$

$M ~ = ~ \text{Measurement}$


## Relative TEM

$$ \% ~ TEM ~ = ~ \frac{TEM}{mean} ~ \times ~ 100 $$

## Total TEM for two observers and two measurements per observer

$$ \text{Total TEM} ~ = ~ \sqrt{\frac{TEM ~ \times ~ (intra_1) ^ 2 ~ + ~ TEM ~ \times ~ (intra_2) ^ 2}{2} ~ + ~ TEM \times (inter) ^ 2} $$

## Total TEM for three observers and two measurements per observer

$$ \text{Total TEM} ~ = ~ \sqrt{\frac{TEM ~ \times ~ (intra_1) ^ 2 ~ + ~ TEM ~ \times ~ (intra_2) ^ 2 ~ + ~ TEM ~ \times ~ (intra_3) ^ 2}{3} ~ + ~ TEM ~ \times ~ inter ^ 2}  $$

## Relative total TEM

$$ \% ~ \text{Total TEM} ~ =  ~ \frac{\text{Total TEM}}{mean} ~ \times ~ 100 $$


## Coefficient of Reliability (R)

$$ R ~ = ~ 1 ~ - ~ \frac{\text{Total TEM} ^ 2}{SD ^ 2} $$