Calculates the regression model (of principal components) analysis of variance (ANOVA) values.

## Syntax

**PCR_ANOVA** (**X**, Mask, **Y**, Intercept, **Return**)

**X**- is the independent variables data matrix, such that each column represents one variable.
**Mask**- is the boolean array to choose the explanatory variables in the model. If missing, all variables in X are included.
**Y**- is the response or the dependent variable data array (a one-dimensional array of cells (e.g., rows or columns)).
**Intercept**- is the constant or the intercept value to fix (e.g., zero). If missing, an intercept will not be fixed and is computed normally.
**Return**- is a switch to select the return output (1 = SSR (default), 2 = SSE, 3 = SST, 4 = MSR, 5 = MSE, 6 = F-Stat, 7 = P-Value).
Value Return 1 SSR ( **default**).2 SSE. 3 SST. 4 MSR. 5 MSE. 6 F-Stat. 7 P-Value.

## Remarks

- The underlying model is described here.
- $\mathbf{y} = \alpha + \beta_1 \times \mathbf{PC}_1 + \dots + \beta_p \times \mathbf{PC}_p$
- The regression ANOVA table examines the following hypothesis: $$\mathbf{H}_o: \beta_1 = \beta_2 = \dots = \beta_p = 0$$ $$\mathbf{H}_1: \exists \beta_i \neq 0, i \in \left[1,0 \right ]$$
- In other words, the regression ANOVA examines the probability that the regression
**does NOT explain**the variation in $\mathbf{y}$, i.e., that any fit is__due purely to chance__. - The MLR_ANOVA calculates the different values in the ANOVA tables as follows: $$\mathbf{SST}=\sum_{i=1}^N \left(Y_i - \bar Y \right )^2$$ $$\mathbf{SSR}=\sum_{i=1}^N \left(\hat Y_i - \bar Y \right )^2$$ $$\mathbf{SSE}=\sum_{i=1}^N \left(Y_i - \hat Y_i \right )^2$$ Where:
- $\mathbf{PC}$ is the principal component.
- $N$ is the number of non-missing observations in the sample data.
- $\bar Y$ is the empirical sample average for the dependent variable.
- $\hat Y_i$ is the regression model estimate value for the i-th observation.
- $\mathbf{SST}$ is the total sum of squares for the dependent variable.
- $\mathbf{SSR}$ is the total sum of squares for the regression (i.e., $\hat y$) estimate.
- $\mathbf{SSE}$ is the total sum of error (aka residuals $\epsilon$) terms for the regression (i.e., $\epsilon = y - \hat y$) estimate.
- $\mathbf{SST} = \mathbf{SSR} + \mathbf{SSE}$

- $p$ is the number of explanatory (aka predictor) variables in the regression.
- $\mathbf{MSR}$ is the mean squares of the regression.
- $\mathbf{MSE}$ is the mean squares of the residuals.
- $\textrm{F-Stat}$ is the test score of the hypothesis.
- $\textrm{F-Stat} \sim \mathbf{F}\left(p,N-p-1 \right)$.

- The sample data may include missing values.
- Each column in the input matrix corresponds to a separate variable.
- Each row in the input matrix corresponds to an observation.
- Observations (i.e., rows) with missing values in X or Y are removed.
- The number of rows of the response variable (Y) must be equal to the number of rows of the explanatory variable (X).
- The MLR_ANOVA function is available starting with version 1.60 APACHE.

## Files Examples

## Related Links

## References

- Hamilton, J.D.; Time Series Analysis, Princeton University Press (1994), ISBN 0-691-04289-6.
- Kenney, J. F. and Keeping, E. S. (1962) "Linear Regression and Correlation." Ch. 15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 252-285.

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