# anova

Analysis of variance for linear regression model

## Syntax

``tbl = anova(mdl)``
``tbl = anova(mdl,anovatype)``
``tbl = anova(mdl,'component',sstype)``

## Description

example

````tbl = anova(mdl)` returns a table with component ANOVA statistics.```

example

````tbl = anova(mdl,anovatype)` returns ANOVA statistics of the specified type `anovatype`. For example, specify `anovatype` as `'component'`(default) to return a table with component ANOVA statistics, or specify `anovatype` as `'summary'` to return a table with summary ANOVA statistics.```
````tbl = anova(mdl,'component',sstype)` computes component ANOVA statistics using the specified type of sum of squares.```

## Examples

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Create a component ANOVA table from a linear regression model of the `hospital` data set.

Load the `hospital` data set and create a model of blood pressure as a function of age and gender.

```load hospital tbl = table(hospital.Age,hospital.Sex,hospital.BloodPressure(:,2), ... 'VariableNames',{'Age','Sex','BloodPressure'}); tbl.Sex = categorical(tbl.Sex); mdl = fitlm(tbl,'BloodPressure ~ Sex + Age^2')```
```mdl = Linear regression model: BloodPressure ~ 1 + Age + Sex + Age^2 Estimated Coefficients: Estimate SE tStat pValue _________ ________ ________ _________ (Intercept) 63.942 19.194 3.3314 0.0012275 Age 0.90673 1.0442 0.86837 0.38736 Sex_Male 3.0019 1.3765 2.1808 0.031643 Age^2 -0.011275 0.013853 -0.81389 0.41772 Number of observations: 100, Error degrees of freedom: 96 Root Mean Squared Error: 6.83 R-squared: 0.0577, Adjusted R-Squared: 0.0283 F-statistic vs. constant model: 1.96, p-value = 0.125 ```

Create an ANOVA table of the model.

`tbl = anova(mdl)`
```tbl=4×5 table SumSq DF MeanSq F pValue ______ __ ______ _______ ________ Age 18.705 1 18.705 0.40055 0.52831 Sex 222.09 1 222.09 4.7558 0.031643 Age^2 30.934 1 30.934 0.66242 0.41772 Error 4483.1 96 46.699 ```

The table displays the following columns for each term except the constant (intercept) term:

• `SumSq` — Sum of squares explained by the term.

• `DF` — Degrees of freedom. In this example, `DF` is 1 for each term in the model and n p for the error term, where n is the number of observations and p is the number of coefficients (including the intercept) in the model. For example, the `DF` for the error term in this model is 100 – 4 = 96. If any variable in the model is a categorical variable, the `DF` for that variable is the number of indicator variables created for its categories (number of categories – 1).

• `MeanSq` — Mean square, defined by `MeanSq = SumSq/DF`. For example, the mean square of the error term, mean squared error (MSE), is 4.4831e+03/96 = 46.6991.

• `F`F-statistic value to test the null hypothesis that the corresponding coefficient is zero, computed by `F = MeanSq/MSE`, where `MSE` is the mean squared error. When the null hypothesis is true, the F-statistic follows the F-distribution. The numerator degrees of freedom is the `DF` value for the corresponding term, and the denominator degrees of freedom is n p. In this example, each F-statistic follows an ${F}_{\left(1,96\right)}$-distribution.

• `pValue`p-value of the F-statistic value. For example, the p-value for `Age` is 0.5283, implying that `Age` is not significant at the 5% significance level given the other terms in the model.

Create a summary ANOVA table from a linear regression model of the `hospital` data set.

Load the `hospital` data set and create a model of blood pressure as a function of age and gender.

```load hospital tbl = table(hospital.Age,hospital.Sex,hospital.BloodPressure(:,2), ... 'VariableNames',{'Age','Sex','BloodPressure'}); tbl.Sex = categorical(tbl.Sex); mdl = fitlm(tbl,'BloodPressure ~ Sex + Age^2')```
```mdl = Linear regression model: BloodPressure ~ 1 + Age + Sex + Age^2 Estimated Coefficients: Estimate SE tStat pValue _________ ________ ________ _________ (Intercept) 63.942 19.194 3.3314 0.0012275 Age 0.90673 1.0442 0.86837 0.38736 Sex_Male 3.0019 1.3765 2.1808 0.031643 Age^2 -0.011275 0.013853 -0.81389 0.41772 Number of observations: 100, Error degrees of freedom: 96 Root Mean Squared Error: 6.83 R-squared: 0.0577, Adjusted R-Squared: 0.0283 F-statistic vs. constant model: 1.96, p-value = 0.125 ```

Create a summary ANOVA table of the model.

`tbl = anova(mdl,'summary')`
```tbl=7×5 table SumSq DF MeanSq F pValue ______ __ ______ _______ ________ Total 4757.8 99 48.059 Model 274.73 3 91.577 1.961 0.12501 . Linear 243.8 2 121.9 2.6103 0.078726 . Nonlinear 30.934 1 30.934 0.66242 0.41772 Residual 4483.1 96 46.699 . Lack of fit 1483.1 39 38.028 0.72253 0.85732 . Pure error 3000 57 52.632 ```

The table displays tests for groups of terms: `Total`, `Model`, and `Residual`.

• `Total` — This row shows the total sum of squares (`SumSq`), degrees of freedom (`DF`), and the mean squared error (`MeanSq`). Note that `MeanSq = SumSq/DF`.

• `Model` — This row includes `SumSq`, `DF`, `MeanSq`, F-statistic value (`F`), and p-value (`pValue`). Because this model includes a nonlinear term (`Age^2`), `anova` partitions the sum of squares (`SumSq`) of `Model` into two parts: `SumSq` explained by the linear terms (`Age` and `Sex`) and `SumSq` explained by the nonlinear term (`Age^2`). The corresponding F-statistic values are for testing the significance of the linear terms and the nonlinear term as separate groups. The nonlinear group consists of the `Age^2` term only, so it has the same p-value as the `Age^2` term in the Component ANOVA Table.

• `Residual` — This row includes `SumSq`, `DF`, `MeanSq`, `F`, and `pValue`. Because the data set includes replications, `anova` partitions the residual `SumSq` into the part for the replications (`Pure error`) and the rest (`Lack of fit`). To test the lack of fit, `anova` computes the F-statistic value by comparing the model residuals to the model-free variance estimate computed on the replications. The F-statistic value shows no evidence of lack of fit.

Fit a linear regression model that contains a categorical predictor. Reorder the categories of the categorical predictor to control the reference level in the model. Then, use `anova` to test the significance of the categorical variable.

Model with Categorical Predictor

Load the `carsmall` data set and create a linear regression model of `MPG` as a function of `Model_Year`. To treat the numeric vector `Model_Year` as a categorical variable, identify the predictor using the `'CategoricalVars'` name-value pair argument.

```load carsmall mdl = fitlm(Model_Year,MPG,'CategoricalVars',1,'VarNames',{'Model_Year','MPG'})```
```mdl = Linear regression model: MPG ~ 1 + Model_Year Estimated Coefficients: Estimate SE tStat pValue ________ ______ ______ __________ (Intercept) 17.69 1.0328 17.127 3.2371e-30 Model_Year_76 3.8839 1.4059 2.7625 0.0069402 Model_Year_82 14.02 1.4369 9.7571 8.2164e-16 Number of observations: 94, Error degrees of freedom: 91 Root Mean Squared Error: 5.56 R-squared: 0.531, Adjusted R-Squared: 0.521 F-statistic vs. constant model: 51.6, p-value = 1.07e-15 ```

The model formula in the display, `MPG ~ 1 + Model_Year`, corresponds to

$\mathrm{MPG}={\beta }_{0}+{\beta }_{1}{Ι}_{\mathrm{Year}=76}+{\beta }_{2}{Ι}_{\mathrm{Year}=82}+ϵ$,

where ${Ι}_{\mathrm{Year}=76}$ and ${Ι}_{\mathrm{Year}=82}$ are indicator variables whose value is one if the value of `Model_Year` is 76 and 82, respectively. The `Model_Year` variable includes three distinct values, which you can check by using the `unique` function.

`unique(Model_Year)`
```ans = 3×1 70 76 82 ```

`fitlm` chooses the smallest value in `Model_Year` as a reference level (`'70'`) and creates two indicator variables ${Ι}_{\mathrm{Year}=76}$ and ${Ι}_{\mathrm{Year}=82}$. The model includes only two indicator variables because the design matrix becomes rank deficient if the model includes three indicator variables (one for each level) and an intercept term.

Model with Full Indicator Variables

You can interpret the model formula of `mdl` as a model that has three indicator variables without an intercept term:

$\mathit{y}={\beta }_{0}{Ι}_{{\mathit{x}}_{1}=70}+\left({\beta }_{0}+{\beta }_{1}\right){Ι}_{{\mathit{x}}_{1}=76}+\left({{\beta }_{0}+\beta }_{2}\right){Ι}_{{\mathit{x}}_{2}=82}+ϵ$.

Alternatively, you can create a model that has three indicator variables without an intercept term by manually creating indicator variables and specifying the model formula.

```temp_Year = dummyvar(categorical(Model_Year)); Model_Year_70 = temp_Year(:,1); Model_Year_76 = temp_Year(:,2); Model_Year_82 = temp_Year(:,3); tbl = table(Model_Year_70,Model_Year_76,Model_Year_82,MPG); mdl = fitlm(tbl,'MPG ~ Model_Year_70 + Model_Year_76 + Model_Year_82 - 1')```
```mdl = Linear regression model: MPG ~ Model_Year_70 + Model_Year_76 + Model_Year_82 Estimated Coefficients: Estimate SE tStat pValue ________ _______ ______ __________ Model_Year_70 17.69 1.0328 17.127 3.2371e-30 Model_Year_76 21.574 0.95387 22.617 4.0156e-39 Model_Year_82 31.71 0.99896 31.743 5.2234e-51 Number of observations: 94, Error degrees of freedom: 91 Root Mean Squared Error: 5.56 ```

Choose Reference Level in Model

You can choose a reference level by modifying the order of categories in a categorical variable. First, create a categorical variable `Year`.

`Year = categorical(Model_Year);`

Check the order of categories by using the `categories` function.

`categories(Year)`
```ans = 3x1 cell {'70'} {'76'} {'82'} ```

If you use `Year` as a predictor variable, then `fitlm` chooses the first category `'70'` as a reference level. Reorder `Year` by using the `reordercats` function.

```Year_reordered = reordercats(Year,{'76','70','82'}); categories(Year_reordered)```
```ans = 3x1 cell {'76'} {'70'} {'82'} ```

The first category of `Year_reordered` is `'76'`. Create a linear regression model of `MPG` as a function of `Year_reordered`.

`mdl2 = fitlm(Year_reordered,MPG,'VarNames',{'Model_Year','MPG'})`
```mdl2 = Linear regression model: MPG ~ 1 + Model_Year Estimated Coefficients: Estimate SE tStat pValue ________ _______ _______ __________ (Intercept) 21.574 0.95387 22.617 4.0156e-39 Model_Year_70 -3.8839 1.4059 -2.7625 0.0069402 Model_Year_82 10.136 1.3812 7.3385 8.7634e-11 Number of observations: 94, Error degrees of freedom: 91 Root Mean Squared Error: 5.56 R-squared: 0.531, Adjusted R-Squared: 0.521 F-statistic vs. constant model: 51.6, p-value = 1.07e-15 ```

`mdl2` uses `'76'` as a reference level and includes two indicator variables ${Ι}_{\mathrm{Year}=70}$ and ${Ι}_{\mathrm{Year}=82}$.

Evaluate Categorical Predictor

The model display of `mdl2` includes a p-value of each term to test whether or not the corresponding coefficient is equal to zero. Each p-value examines each indicator variable. To examine the categorical variable `Model_Year` as a group of indicator variables, use `anova`. Use the `'components'`(default) option to return a component ANOVA table that includes ANOVA statistics for each variable in the model except the constant term.

`anova(mdl2,'components')`
```ans=2×5 table SumSq DF MeanSq F pValue ______ __ ______ _____ __________ Model_Year 3190.1 2 1595.1 51.56 1.0694e-15 Error 2815.2 91 30.936 ```

The component ANOVA table includes the p-value of the `Model_Year` variable, which is smaller than the p-values of the indicator variables.

## Input Arguments

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Linear regression model object, specified as a `LinearModel` object created by using `fitlm` or `stepwiselm`, or a `CompactLinearModel` object created by using `compact`.

ANOVA type, specified as one of these values:

• `'component'``anova` returns the table `tbl` with ANOVA statistics for each variable in the model except the constant term.

• `'summary'``anova` returns the table `tbl` with summary ANOVA statistics for grouped variables and the model as a whole.

For details, see the `tbl` output argument description.

Sum of squares type for each term, specified as one of the values in this table.

ValueDescription
`1`Type 1 sum of squares — Reduction in residual sum of squares obtained by adding the term to a fit that already includes the preceding terms
`2`Type 2 sum of squares — Reduction in residual sum of squares obtained by adding the term to a model that contains all other terms
`3`Type 3 sum of squares — Reduction in residual sum of squares obtained by adding the term to a model that contains all other terms, but with their effects constrained to obey the usual “sigma restrictions” that make models estimable
`'h'`Hierarchical model — Similar to Type 2, but uses both continuous and categorical factors to determine the hierarchy of terms

The sum of squares for any term is determined by comparing two models. For a model containing main effects but no interactions, the value of `sstype` influences the computations on unbalanced data only.

Suppose you are fitting a model with two factors and their interaction, and the terms appear in the order A, B, AB. Let R(·) represent the residual sum of squares for the model. So, R(A, B, AB) is the residual sum of squares fitting the whole model, R(A) is the residual sum of squares fitting the main effect of A only, and R(1) is the residual sum of squares fitting the mean only. The three sum of squares types are as follows:

TermType 1 Sum of SquaresType 2 Sum of SquaresType 3 Sum of Squares

A

R(1) – R(A)

R(B) – R(A, B)

R(B, AB) – R(A, B, AB)

B

R(A) – R(A, B)

R(A) – R(A, B)

R(A, AB) – R(A, B, AB)

AB

R(A, B) – R(A, B, AB)

R(A, B) – R(A, B, AB)

R(A, B) – R(A, B, AB)

The models for Type 3 sum of squares have sigma restrictions imposed. This means, for example, that in fitting R(B, AB), the array of AB effects is constrained to sum to 0 over A for each value of B, and over B for each value of A.

For Type 3 sum of squares:

• If `mdl` is a `CompactLinearModel` object and the regression model is nonhierarchical, `anova` returns an error.

• If `mdl` is a `LinearModel` object and the regression model is nonhierarchical, `anova` refits the model using effects coding whenever it needs to compute a Type 3 sum of squares.

• If the regression model in `mdl` is hierarchical, `anova` computes the results without refitting the model.

`sstype` applies only if `anovatype` is `'component'`.

## Output Arguments

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ANOVA summary statistics table, returned as a table.

The contents of `tbl` depend on the ANOVA type specified in `anovatype`.

• If `anovatype` is `'component'`, then `tbl` contains ANOVA statistics for each variable in the model except the constant (intercept) term. The table includes these columns for each variable:

ColumnDescription
`SumSq`

Sum of squares explained by the term, computed depending on `sstype`

`DF`

Degrees of freedom

• `DF` of a numeric variable is 1.

• `DF` of a categorical variable is the number of indicator variables created for the category (number of categories – 1). Note that `tbl` contains one row for each categorical variable instead of one row for each indicator variable as in the model display. Use `anova` to test a categorical variable as a group of indicator variables.

• `DF` of an error term is n – p, where n is the number of observations and p is the number of coefficients in the model.

`MeanSq`

Mean square, defined by `MeanSq` = `SumSq`/`DF`

`MeanSq` for the error term is the mean squared error (MSE).

`F`

F-statistic value to test the null hypothesis that the corresponding coefficient is zero, computed by `F` = `MeanSq`/`MSE`

When the null hypothesis is true, the F-statistic follows the F-distribution. The numerator degrees of freedom is the `DF` value for the corresponding term, and the denominator degrees of freedom is n – p.

`pValue`

p-value of the F-statistic value

For an example, see Component ANOVA Table.

• If `anovatype` is `'summary'`, then `tbl` contains summary statistics of grouped terms for each row. The table includes the same columns as `'component'` and these rows:

RowDescription
`Total`

Total statistics

• `SumSq` — Total sum of squares, which is the sum of the squared deviations of the response around its mean

• `DF` — Sum of degrees of freedom of `Model` and `Residual`

`Model`

Statistics for the model as a whole

• `SumSq` — Model sum of squares, which is the sum of the squared deviations of the fitted value around the response mean.

• `F` and `pValue` — These values provide a test of whether the model as a whole fits significantly better than a degenerate model consisting of only a constant term.

If `mdl` includes only linear terms, then `anova` does not decompose `Model` into `Linear` and `NonLinear`.

`Linear`

Statistics for linear terms

• `SumSq` — Sum of squares for linear terms, which is the difference between the model sum of squares and the sum of squares for nonlinear terms.

• `F` and `pValue` — These values provide a test of whether the model with only linear terms fits better than a degenerate model consisting of only a constant term. `anova` uses the mean squared error that is based on the full model to compute this F-value, so the F-value obtained by dropping the nonlinear terms and repeating the test is not the same as the value in this row.

`Nonlinear`

Statistics for nonlinear terms

• `SumSq` — Sum of squares for nonlinear (higher-order or interaction) terms, which is the increase in the residual sum of squares obtained by keeping only the linear terms and dropping all nonlinear terms.

• `F` and `pValue` — These values provide a test of whether the full model fits significantly better than a smaller model consisting of only the linear terms.

`Residual`

Statistics for residuals

• `SumSq` — Residual sum of squares, which is the sum of the squared residual values

• `MeanSq` — Mean squared error, used to compute the F-statistic values for `Model`, `Linear`, and `NonLinear`

If `mdl` is a full `LinearModel` object and the sample data contains replications (multiple observations sharing the same predictor values), then `anova` decomposes the residual sum of squares into a sum of squares for the replicated observations (```Lack of fit```) and the remaining sum of squares (`Pure error`).

`Lack of fit`

Lack-of-fit statistics

• `SumSq` — Sum of squares due to lack of fit, which is the difference between the residual sum of squares and the replication sum of squares.

• `F` and `pValue` — The F-statistic value is the ratio of lack-of-fit `MeanSq` to pure error `MeanSq`. The ratio provides a test of bias by measuring whether the variation of the residuals is larger than the variation of the replications. A low p-value implies that adding additional terms to the model can improve the fit.

`Pure error`

Statistics for pure error

• `SumSq` — Replication sum of squares, obtained by finding the sets of points with identical predictor values, computing the sum of squared deviations around the mean within each set, and pooling the computed values

• `MeanSq` — Model-free pure error variance estimate of the response

For an example, see Summary ANOVA Table.

## Alternative Functionality

More complete ANOVA statistics are available in the `anova1`, `anova2`, and `anovan` functions.