## F-statistic and t-statistic

### F-statistic

#### Purpose

In linear regression, the F-statistic is the test statistic for the analysis of variance (ANOVA) approach to test the significance of the model or the components in the model.

#### Definition

The F-statistic in the linear model output display is the test statistic for testing the statistical significance of the model. The F-statistic values in the anova display are for assessing the significance of the terms or components in the model.

#### How To

After obtaining a fitted model, say, mdl, using fitlm or stepwiselm, you can:

• Find the F-statistic vs. constant model in the output display or by using

disp(mdl)
• Display the ANOVA for the model using

anova(mdl,'summary')
• Obtain the F-statistic values for the components, except for the constant term using

anova(mdl)
For details, see the anova method of the LinearModel class.

### Assess Fit of Model Using F-statistic

This example shows how to assess the fit of the model and the significance of the regression coefficients using the F-statistic.

tbl = table(hospital.Age,hospital.Weight,hospital.Smoker,hospital.BloodPressure(:,1), ...
'VariableNames',{'Age','Weight','Smoker','BloodPressure'});
tbl.Smoker = categorical(tbl.Smoker);

Fit a linear regression model.

mdl = fitlm(tbl,'BloodPressure ~ Age*Weight + Smoker + Weight^2')
mdl =
Linear regression model:
BloodPressure ~ 1 + Smoker + Age*Weight + Weight^2

Estimated Coefficients:
Estimate        SE         tStat        pValue
__________    _________    ________    __________

(Intercept)        168.02       27.694       6.067    2.7149e-08
Age              0.079569      0.39861     0.19962       0.84221
Weight           -0.69041       0.3435     -2.0099      0.047305
Smoker_true        9.8027       1.0256      9.5584    1.5969e-15
Age:Weight     0.00021796    0.0025258    0.086294       0.93142
Weight^2        0.0021877    0.0011037      1.9822      0.050375

Number of observations: 100, Error degrees of freedom: 94
Root Mean Squared Error: 4.73
F-statistic vs. constant model: 21, p-value = 4.81e-14

The F-statistic of the linear fit versus the constant model is 21, with a p-value of 4.81e-14. The model is significant at the 5% significance level. The R-squared value of 0.528 means the model explains about 53% of the variability in the response. There might be other predictor (explanatory) variables that are not included in the current model.

Display the ANOVA table for the fitted model.

anova(mdl,'summary')
ans=5×5 table
SumSq     DF    MeanSq      F         pValue
______    __    ______    ______    __________

Total          4461.2    99    45.062
Model          2354.5     5     470.9    21.012    4.8099e-14
. Linear       2263.3     3    754.42    33.663    7.2417e-15
. Nonlinear    91.248     2    45.624    2.0358        0.1363
Residual       2106.6    94    22.411

This display separates the variability in the model into linear and nonlinear terms. Since there are two non-linear terms (Weight^2 and the interaction between Weight and Age), the nonlinear degrees of freedom in the DF column is 2. There are three linear terms in the model (one Smoker indicator variable, Weight, and Age). The corresponding F-statistics in the F column are for testing the significance of the linear and nonlinear terms as separate groups.

When there are replicated observations, the residual term is also separated into two parts; first is the error due to the lack of fit, and second is the pure error independent from the model, obtained from the replicated observations. In that case, the F-statistic is for testing the lack of fit, that is, whether the fit is adequate or not. But, in this example, there are no replicated observations.

Display the ANOVA table for the model terms.

anova(mdl)
ans=6×5 table
SumSq      DF     MeanSq         F          pValue
________    __    ________    _________    __________

Age             62.991     1      62.991       2.8107      0.096959
Weight        0.064104     1    0.064104    0.0028604       0.95746
Smoker          2047.5     1      2047.5       91.363    1.5969e-15
Age:Weight     0.16689     1     0.16689    0.0074466       0.93142
Weight^2        88.057     1      88.057       3.9292      0.050375
Error           2106.6    94      22.411

This display decomposes the ANOVA table into the model terms. The corresponding F-statistics in the F column assess the statistical significance of each term. For example, the F-test for Smoker tests whether the coefficient of the indicator variable for Smoker is different from zero. That is, the F-test determines whether being a smoker has a significant effect on BloodPressure. The degrees of freedom for each model term is the numerator degrees of freedom for the corresponding F-test. All the terms have one degree of freedom. In the case of a categorical variable, the degrees of freedom is the number of indicator variables. Smoker has only one indicator variable, so it also has one degree of freedom.

### t-statistic

#### Purpose

In linear regression, the t-statistic is useful for making inferences about the regression coefficients. The hypothesis test on coefficient i tests the null hypothesis that it is equal to zero – meaning the corresponding term is not significant – versus the alternate hypothesis that the coefficient is different from zero.

#### Definition

For a hypotheses test on coefficient i, with

H0 : βi = 0

H1 : βi ≠ 0,

the t-statistic is:

$t=\frac{{b}_{i}}{SE\left({b}_{i}\right)},$

where SE(bi) is the standard error of the estimated coefficient bi.

#### How To

After obtaining a fitted model, say, mdl, using fitlm or stepwiselm, you can:

• Find the coefficient estimates, the standard errors of the estimates (SE), and the t-statistic values of hypothesis tests for the corresponding coefficients (tStat) in the output display.

• Call for the display using

display(mdl)

### Assess Significance of Regression Coefficients Using t-statistic

This example shows how to test for the significance of the regression coefficients using t-statistic.

Load the sample data and fit the linear regression model.

mdl = fitlm(ingredients,heat)
mdl =
Linear regression model:
y ~ 1 + x1 + x2 + x3 + x4

Estimated Coefficients:
Estimate      SE        tStat       pValue
________    _______    ________    ________

(Intercept)      62.405     70.071      0.8906     0.39913
x1               1.5511    0.74477      2.0827    0.070822
x2              0.51017    0.72379     0.70486      0.5009
x3              0.10191    0.75471     0.13503     0.89592
x4             -0.14406    0.70905    -0.20317     0.84407

Number of observations: 13, Error degrees of freedom: 8
Root Mean Squared Error: 2.45
F-statistic vs. constant model: 111, p-value = 4.76e-07

You can see that for each coefficient, tStat = Estimate/SE. The $p$-values for the hypotheses tests are in the pValue column. Each $t$-statistic tests for the significance of each term given other terms in the model. According to these results, none of the coefficients seem significant at the 5% significance level, although the R-squared value for the model is really high at 0.97. This often indicates possible multicollinearity among the predictor variables.

Use stepwise regression to decide which variables to include in the model.

mdl = stepwiselm(ingredients,heat)
1. Adding x4, FStat = 22.7985, pValue = 0.000576232
2. Adding x1, FStat = 108.2239, pValue = 1.105281e-06
mdl =
Linear regression model:
y ~ 1 + x1 + x4

Estimated Coefficients:
Estimate       SE        tStat       pValue
________    ________    _______    __________

(Intercept)       103.1       2.124      48.54    3.3243e-13
x1                 1.44     0.13842     10.403    1.1053e-06
x4             -0.61395    0.048645    -12.621    1.8149e-07

Number of observations: 13, Error degrees of freedom: 10
Root Mean Squared Error: 2.73
F-statistic vs. constant model: 177, p-value = 1.58e-08

In this example, stepwiselm starts with the constant model (default) and uses forward selection to incrementally add x4 and x1. Each predictor variable in the final model is significant given the other one is in the model. The algorithm stops when adding none of the other predictor variables significantly improves in the model. For details on stepwise regression, see stepwiselm.