# The gof test is so important that stata always prints

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The GOF-test is so important that Stata always prints it out Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 14 / 54

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Inference, goodness of fit and modification indices Note that when Σ is completely free, it is a symmetric p × p -matrix, and hence has p × ( p + 1 ) / 2 non-redundant elements. To find the degrees of freedom: Count the number of parameters in the model, and subtract this from p × ( p + 1 ) / 2 Here: p = 6 (the number of observed grades per person) and we counted 13 parameters in total. Hence df = 6 × 7 / 2 - 13 = 8, as reported by Stata. Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 15 / 54
Inference, goodness of fit and modification indices If the goodness-of-fit test is rejected, Stata can compute how much Chi-Square would drop if each fixed parameter is freed. This drop is the parameter’s Modification Index. For example, Verbal’s connection with Math, given by α 1 , 2 , is fixed to zero. Here, the GOF-test is not rejected (the p-value is high, indicating little evidence against the claim that our model is good). Note that the GOF-test only tests the quality of approximation for the covariance matrix – not any other assumptions! Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 16 / 54

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Contents 1 Confirmatory factor analysis Stata: Inputting data based on empirical covariance matrices Inference, goodness of fit and modification indices An empirical example: Mental abilities 2 More on Path models Indirect and direct effects A brief empirical example from sociology On R 2 Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 17 / 54
An empirical example: Mental abilities Consider a psychological study, designed to assess the correlation between school children’s mental abilities in 1 Visual perception 2 Verbal ability 3 Speed in numerics The researchers administered three tests on each of the three mental abilities. A total of 145 seventh- and eight-grade children from the same school took all the nine tests. The tests were i. Visual Perception (visual) ii. Cubes (visual) iii. Lozenges (visual) iv. Paragraph Comprehension (verbal) v. Sentence Completion (verbal) vi. Word Meaning (verbal) vii. Addition (speed in numerics) viii. Counting Dots (speed in numerics) ix. Straight-Curved Capitals (speed in numerics) Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 18 / 54

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An empirical example: Mental abilities The researchers’ Confirmatory Factor Model: Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 19 / 54
An empirical example: Mental abilities The path diagram can, just like the simpler grades example, be written in equation form . The latent variables of interest are ξ 1 = Visual ability ξ 2 = Verbal ability ξ 3 = Speed in numerics and the CFA model is X 1 = α 1 , 1 ξ 1 + δ 1 X 2 = α 2 , 1 ξ 1 + δ 2 X 3 = α 3 , 1 ξ 1 + δ 3 X 4 = α 4 , 2 ξ 2 + δ 4 X 5 = α 5 , 2 ξ 2 + δ 5 X 6 = α 6 , 2 ξ 2 + δ 6 X 7 = α 7 , 3 ξ 3 + δ 7 X 8 = α 8 , 3 ξ 3 + δ 8 X 9 = α 9 , 3 ξ 3 + δ 9 ξ 1 , ξ 2 , ξ 3 are allowed to correlate freely.

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