We're almost there! We just need to determine an objective way of deciding when too much of the error in our prediction is due to lack of model fit. That's where the lack of fit *F*-test comes into play. Let's return to the first checking account example,(New Accounts data):

Jumping ahead to the punchline, here's Minitab's output for the lack of fit *F*-test for this data set:

##### Analysis of Variance

Source | DF | Adj SS | Adj MS | F-Value | P-Value |
---|---|---|---|---|---|

Regression | 1 | 5141 | 5141 | 3.14 | 0.110 |

Residual Error | 9 | 14742 | 1638 | ||

Lack of Fit | 4 | 13594 | 3398 | 14.80 | 0.006 |

Pure Error | 5 | 1148 | 230 | ||

Total | 10 | 19883 |

1 rows with no replicates

As you can see, the lack of fit output appears as a portion of the analysis of variance table. In the **Sum of Squares** ("**SS**") column, we see — as we previously calculated — that *SSLF* = 13594 and *SSPE* = 1148 sum to *SSE* = 14742. We also see in the **Degrees of Freedom** ("**DF**") column that — since there are *n* = 11 data points and *c* = 6 distinct *x* values (75, 100, 125, 150, 175, and 200) — the lack of fit degrees of freedom *c *- 2 = 4 and the pure error degrees of freedom is *n* -* c* = 5 sum to the error degrees of freedom *n* - 2 = 9.

Just as is done for the sums of squares in the basic analysis of variance table, the lack of fit sum of squares and the error sum of squares are used to calculate "mean squares." They are even calculated similarly, namely by dividing the sum of squares by its associated degrees of freedom. Here are the formal definitions of the mean squares:

**lack of fit mean square**" is \(MSLF=\dfrac{\sum\sum(\bar{y}_i-\hat{y}_{ij})^2}{c-2}=\dfrac{SSLF}{c-2}\)

**pure error mean square**" is \(MSPE=\dfrac{\sum\sum(y_{ij}-\bar{y}_{i})^2}{n-c}=\dfrac{SSPE}{n-c}\)

In the **Mean Squares** ("* MS*") column, we see that the lack of fit mean square

*MSLF*is 13594 divided by 4, or 3398. The pure error mean square

*MSPE*is 1148 divided by 5, or 230:

##### Analysis of Variance

Source | DF | Adj SS | Adj MS | F-Value | P-Value |
---|---|---|---|---|---|

Regression |
1 | 5141 | 5141 | 3.14 | 0.110 |

Residual Error |
9 | 14742 | 1638 | ||

Lack of Fit |
4 | 13594 | 3398 | 14.80 | 0.006 |

Pure Error |
5 | 1148 | 230 | ||

Total |
10 | 19883 |

You might notice that the lack of fit F-statistic is calculated by dividing the lack of fit mean square (MSLF = 3398) by the pure error mean square (MSPE = 230) to get 14.80. How do we know that this F-statistic helps us in testing the hypotheses:

- \(H_{0 }\): The relationship assumed in the model is reasonable, i.e., there is no lack of fit.
- \(H_{A }\): The relationship assumed in the model is not reasonable, i.e., there is lack of fit.

The answer lies in the "**expected mean squares**." In our sample of *n* = 11 newly opened checking accounts, we obtained *MSLF* = 3398. If we had taken a different random sample of size *n* = 11, we would have obtained a different value for *MSLF*. Theory tells us that the average of all of the possible *MSLF* values we could obtain is:

\(E(MSLF) =\sigma^2+\dfrac{\sum n_i(\mu_i-(\beta_0+\beta_1X_i))^2}{c-2}\)

That is, we should expect *MSLF*, on average, to equal the above quantity — \(\sigma^{2}\) plus another messy-looking term. Think about that messy term. If the null hypothesis is true, *i.e.*, if the relationship between the predictor *x* and the response *y* is linear, then \(\mu_{i} = \beta_{0} + \beta_{1}X_{i}\) and the messy term becomes 0 and goes away. That is, if there is no lack of fit, we should expect the lack of fit mean square *MSLF* to equal \(\sigma^{2}\).

What should we expect *MSPE* to equal? Theory tells us it should, on average, always equal \(\sigma^{2}\):

\(E(MSPE) =\sigma^2\)

Aha — there we go! The logic behind the calculation of the *F*-statistic is now clear:

- If there is a linear relationship between
*x*and*y*, then \(\mu_{i} = \beta_{0} + \beta_{1}X_{i}\). That is, there is no lack of fit in the simple linear regression model. We would expect the ratio*MSLF*/*MSPE*to be close to 1. - If there is not a linear relationship between
*x*and*y*, then \(\mu_{i} ≠ \beta_{0} + \beta_{1}X_{i}\). That is, there is lack of fit in the simple linear regression model. We would expect the ratio*MSLF*/*MSPE*to be large,*i.e.*, a value greater than 1.

So, to conduct the lack of fit test, we calculate the value of the *F*-statistic:

\(F^*=\dfrac{MSLF}{MSPE}\)

and determine if it is large. To decide if it is large, we compare the *F**-statistic to an *F*-distribution with *c *- 2 numerator degrees of freedom and *n* -* c* denominator degrees of freedom.

##
In summary
Section* *

We follow standard hypothesis test procedures in conducting the lack of fit *F*-test. First, we specify the null and alternative hypotheses:

- \(H_{0}\): The relationship assumed in the model is reasonable, i.e., there is no lack of fit in the model \(\mu_{i} = \beta_{0} + \beta_{1}X_{i}\).
- \(H_{A}\): The relationship assumed in the model is not reasonable, i.e., there is lack of fit in the model \(\mu_{i} = \beta_{0} + \beta_{1}X_{i}\).

Second, we calculate the value of the *F*-statistic:

\(F^*=\dfrac{MSLF}{MSPE}\)

To do so, we complete the analysis of variance table using the following formulas.

##### Analysis of Variance

Source | DF | SS | MS | F |
---|---|---|---|---|

Regression | 1 | \(SSR=\sum_{i=1}^{c}\sum_{j=1}^{n_i}(\hat{y}_{ij}-\bar{y})^2\) | \(MSR=\dfrac{SSR}{1}\) | \(F=\dfrac{MSR}{MSE}\) |

Residual Error | n - 2 |
\(SSE=\sum_{i=1}^{c}\sum_{j=1}^{n_i}(y_{ij}-\hat{y}_{ij})^2\) | \(MSE=\dfrac{SSE}{n-2}\) | |

Lack of Fit | c - 2 |
\(SSLF=\sum_{i=1}^{c}\sum_{j=1}^{n_i}(\bar{y}_{i}-\hat{y}_{ij})^2\) | \(MSLF=\dfrac{SSLF}{c-2}\) | \(F^*=\dfrac{MSLF}{MSPE}\) |

Pure Error | n - c |
\(SSPE=\sum_{i=1}^{c}\sum_{j=1}^{n_i}(y_{ij}-\bar{y}_{i})^2\) | \(MSPE=\dfrac{SSPE}{n-c}\) | |

Total | n - 1 |
\(SSTO=\sum_{i=1}^{c}\sum_{j=1}^{n_i}(y_{ij}-\bar{y})^2\) |

In reality, we let statistical software such as Minitab, determine the analysis of variance table for us.

Third, we use the resulting *F**-statistic to calculate the *P*-value. As always, the *P*-value is the answer to the question "how likely is it that we’d get an *F**-statistic as extreme as we did if the null hypothesis were true?" The *P*-value is determined by referring to an *F-*distribution with *c *- 2 numerator degrees of freedom and *n *-* c* denominator degrees of freedom.

Finally, we make a decision:

- If the
*P*-value is smaller than the significance level \(\alpha\), we reject the null hypothesis in favor of the alternative. We conclude "there is sufficient evidence at the \(\alpha\) level to conclude that there is a lack of fit*in the simple linear regression model*." - If the
*P*-value is larger than the significance level \(\alpha\), we fail to reject the null hypothesis. We conclude "there is not enough evidence at the \(\alpha\) level to conclude that there is a lack of fit*in the simple linear regression model*."

For our checking account example:

in which we obtain:

##### Analysis of Variance

Source | DF | Adj SS | Adj MS | F-Value | P-Value |
---|---|---|---|---|---|

Regression | 1 | 5141 | 5141 | 3.14 | 0.110 |

Residual Error | 9 | 14742 | 1638 | ||

Lack of Fit | 4 | 13594 | 3398 | 14.80 | 0.006 |

Pure Error | 5 | 1148 | 230 | ||

Total | 10 | 19883 |

the *F**-statistic is 14.80 and the *P*-value is 0.006. The *P*-value is smaller than the significance level \(\alpha = 0.05\) — we reject the null hypothesis in favor of the alternative. There is sufficient evidence at the \(\alpha = 0.05\) level to conclude that there is a lack of fit *in the simple linear regression model*. In light of the scatterplot, the lack of fit test provides the answer we expected.

##
Try it!

###
The lack of fit test
Section* *

Fill in the missing numbers (??) in the following analysis of variance table resulting from a simple linear regression analysis.

Click on the light bulb

Source | DF | Adj SS | Adj MS | F-Value | P-Value |
---|---|---|---|---|---|

Regression | ?? | 12.597 | ?? | ?? | 0.000 |

Residual Error | ?? | ?? | ?? | ||

Lack of Fit | 3 | ?? | ?? | ?? | ?? |

Pure Error | ?? | 0.157 | ?? | ||

Total | 14 | 15.522 |