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Changed fig 3. image to the correct F distribution and cited the source
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dkpalmers committed Sep 30, 2024
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17 changes: 8 additions & 9 deletions ancova.qmd
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title: "ANCOVA"
---

```{=html}
<script type="text/javascript">
function showhide(id) {
var e = document.getElementById(id);
e.style.display = (e.style.display == 'block') ? 'none' : 'block';
}
</script>
```

```{r}
#| label: setup
#| include: false
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The assumptions for the ANCOVA model are essentially the same as for [CB\[1\]](cb1.qmd), with the addition that the response and the covariate need to have a linear relationship:

| Requirements | Method for Checking | What You Hope to See |
|--------------------|--------------------------|--------------------------|
| Linear relationship between covariate and response | Residual vs. Fitted Plot | No trend or pattern |
| Constant variance across factor levels | Residual vs. Fitted Plot | No wedge or megaphone shape |
| Normally Distributed Residuals | Normal Q-Q plot | Straight line, majority of points in boundaries |
| Independent residuals | Order plot (only if applicable) | No pattern/trend |
| | Familiarity with/critical thinking about the experiment | No potential source for bias |
| Requirements | Method for Checking | What You Hope to See |
|----|----|----|
| Linear relationship between covariate and response | Residual vs. Fitted Plot | No trend or pattern |
| Constant variance across factor levels | Residual vs. Fitted Plot | No wedge or megaphone shape |
| Normally Distributed Residuals | Normal Q-Q plot | Straight line, majority of points in boundaries |
| Independent residuals | Order plot (only if applicable) | No pattern/trend |
| | Familiarity with/critical thinking about the experiment | No potential source for bias |

The following example illustrates how to conduct an ANOVA in R.

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383 changes: 83 additions & 300 deletions docs/Anova_F-test.html
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42 changes: 22 additions & 20 deletions docs/Anova_F-test.qmd
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Expand Up @@ -118,7 +118,9 @@ F = \frac{\text{Variation between factor level means}}{\text{Variation of indivi
$$
:::

The F distribution is defined by 2 values for degrees of freedom. One degrees of freedom value for the variance estimate in the numerator, and another for the variance estimate in the denominator. The p-value for the ANOVA F-test is calculated as the area under the F distribution curve to the right of the F statistic, as shown in @fig-pvalue .
The F distribution is defined by 2 values for degrees of freedom. One degrees of freedom value for the variance estimate in the numerator, and another for the variance estimate in the denominator. The p-value for the ANOVA F-test is calculated as the area under the F distribution curve to the right of the F statistic, as shown in @fig-pvalue [^4].

[^4]: From the F distribution applet (https://homepage.divms.uiowa.edu/~mbognar/applets/f.html) copyright 2021 by Matt Bogner, Department of Statistics and Actuarial Science, University of Iowa

![Example P-value calculation based on a F-statistic of 2.23 and an F distribution with 3 and 16 degrees of freedom](images/p-value_example.jpg){#fig-pvalue}

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With this is mind, we can fill in the F column of the ANOVA table for Treatment Factor.

| Source | df | SS | MS | F | p-value |
|:-----------------|:---:|:---:|:---:|:-------------------------------------------------------------:|:-------:|
| Grand Mean | | | | | |
| Treatment Factor | | | | $\frac{MS_\text{Treatment Factor}}{MS_\text{Residual Error}}$ | |
| Residual Error | | | | | |
| Total | | | | | |
| Source | df | SS | MS | F | p-value |
|:--------|:-------:|:-------:|:-------:|:-----------------------------:|:-------:|
| Grand Mean | | | | | |
| Treatment Factor | | | | $\frac{MS_\text{Treatment Factor}}{MS_\text{Residual Error}}$ | |
| Residual Error | | | | | |
| Total | | | | | |

: Blank ANOVA summary table for an experiment with 1 treatment factor {#tbl-blank-1}

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Therefore, in a mean square calculation, the numerator is the sum of squares and the denominator is the degrees of freedom.
| Source | df | SS | MS | F | p-value |
|:-----------------|:---:|:---:|:---------------------------------------------------------------:|:-------------------------------------------------------------:|:-------:|
| Grand Mean | | | | | |
| Treatment Factor | | | $\frac{SS_\text{Treatment Factor}}{df_\text{Treatment Factor}}$ | $\frac{MS_\text{Treatment Factor}}{MS_\text{Residual Error}}$ | |
| Residual Error | | | $\frac{SS_\text{Residual Error}}{df_\text{Residual Error}}$ | | |
| Total | | | | | |
| Source | df | SS | MS | F | p-value |
|:--------|:-------:|:-------:|:-----------------:|:----------------:|:-------:|
| Grand Mean | | | | | |
| Treatment Factor | | | $\frac{SS_\text{Treatment Factor}}{df_\text{Treatment Factor}}$ | $\frac{MS_\text{Treatment Factor}}{MS_\text{Residual Error}}$ | |
| Residual Error | | | $\frac{SS_\text{Residual Error}}{df_\text{Residual Error}}$ | | |
| Total | | | | | |
## Sum of Squares (SS)
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In the table below an equation for each factor's SS is listed using terms from the factor effects model. We'll walk through the meaning of each of those equations.
| Source | df | SS | MS | F | p-value |
|:-----------------|:---:|:-------------------------------------:|:---------------------------------------------------------------:|:-------------------------------------------------------------:|:-------:|
| Grand Mean | | $n*\bar{y}_\text{..}^2$ | | | |
| Treatment Factor | | $$ \sum (\hat{\alpha}_i^2*n_i)$$ | $\frac{SS_\text{Treatment Factor}}{df_\text{Treatment Factor}}$ | $\frac{MS_\text{Treatment Factor}}{MS_\text{Residual Error}}$ | |
| Residual Error | | $$ \sum \hat{\epsilon}_\text{ij}^2 $$ | $\frac{SS_\text{Residual Error}}{df_\text{Residual Error}}$ | | |
| Total | | $\sum y_\text{ij}^2$ | | | |
| Source | df | SS | MS | F | p-value |
|:----------|:---------:|:---------:|:--------------:|:-------------:|:---------:|
| Grand Mean | | $n*\bar{y}_\text{..}^2$ | | | |
| Treatment Factor | | $$ \sum (\hat{\alpha}_i^2*n_i)$$ | $\frac{SS_\text{Treatment Factor}}{df_\text{Treatment Factor}}$ | $\frac{MS_\text{Treatment Factor}}{MS_\text{Residual Error}}$ | |
| Residual Error | | $$ \sum \hat{\epsilon}_\text{ij}^2 $$ | $\frac{SS_\text{Residual Error}}{df_\text{Residual Error}}$ | | |
| Total | | $\sum y_\text{ij}^2$ | | | |
::: column-margin
A deviation from the mean can be thought of as an effect. That is why the symbols for factor effects are used in the SS column in the ANOVA summary table.
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The example described above is summarized in @tbl-df-example. The first number is represented as an $a$ and the second number if represented with a $b$.
| value 1 | value 2 | value 3 | | Mean of 3 Values |
|:------------:|:------------:|:---------------------------:|:---:|:----------------:|
|:-----------:|:-----------:|:-------------------:|:-----------:|:-----------:|
| a | b | $3*10 - (a+b)$ | -\> | 10 |
| free to vary | free to vary | depends on other two values | | |
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