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Applied Linear Statistical Models

Diagnostic Methods - F Tests for Lack of Fit Dr. DH Jones

Example: Bank Data

X is the minimal deposit to receive gift, Y the number of new savings accounts opened. Each case represents the experience at unique bank branches. Scatter plot of data clearly shows that the simple linear model does not fit the data. Note the replications at most of the X's.

Scatterplot

ANOVA Table

ANOVA a

Model 1

Regression Residual Total

Sum of Squares 5141.34 14741.6 19882.9

df 1 9 10

Mean Square 5141.34 1637.95

F 3.139

Sig. .110b

a. Dependent Variable: ACCOUNTS b. Independent Variables: (Constant), DEPOSIT

Formula for SSE with Replications

Yij represents the ith case of the jth replication group SSE = Yij - Yij

(

)

2

Note: Fitted values of the same jth replication group are all the same

Components of the Sum of Squares for Error

Yij - Yij = Yij - Y j + Y j - Yij

(Y

(

ij

- Yij

) = (Y

2

) (

)

- Yj

ij

) + (Y

2

j

- Yij

)

2

SSE = SSPE + SSLF n - 2 = ( n - c) + ( c - 2)

Note: The Sum of Squares for Pure Error SSPE is also called the "Within Groups Sum of Squares." Note: The Sum of Squares for Lack of Fit SSLF is also called the "Sum of Squares Deviation from Linearity."

Example: Box Plot of Bank Data Residuals from Fitted Line with Group Means.

Expected Values of Mean Squares

E{ MSPE} = 2 E{ MSLF } = 2 +

n j µ j - 0 + 1 X j

c-2

[ (

)]

2

Example: Bank Data ANOVA Table

ANOVA Table

ACCOUNTS * DEPOSIT

Between Groups

(Combined) Linearity Deviation from Linearity

Sum of Squares 18734.9 5141.34 13593.6 1148.00 19882.9

df 5 1 4 5 10

Mean Square 3746.98 5141.34 3398.39 229.600

F 16.320 22.393 14.801

Sig. .004 .005 .006

Within Groups Total

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Applied Linear Statistical Models

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