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Model Modi cation There are at least two reasons for modifying a SEM model: to test hypotheses (in theoretical work) and to improve t (especially in exploratory work) Structural equation modeling is a con rmatory technique; therefore, when model modi cation is done to improve t, the analysis changes from con rmatory to exploratory Any conclusions drawn from a model that has undergone substantial modi cation should be viewed extremely cautiously Cross validation should be performed on modi ed models whenever possible The three basic methods of model modi cation are the chi-square difference, Lagrange multiplier (LM), and Wald tests All are asymptotically equivalent under the null hypothesis but approach model modi cation differently In this section each of these approaches is discussed with reference to the Acceptance of Risky Behavior example, and examples will be shown where relevant Chi-Square Difference Test If models are nested (ie.

1. 2. 3. 4. 5.

, models are subsets of each other), the chi-square value for the larger model is subtracted from the chi-square value for the smaller, nested model, and the difference, also a chi-square, is evaluated with degrees of freedom equal to the difference between the degrees of freedom in the two models In the Acceptance of Risky Behavior model we could test whether gender predicted Acceptance of Risky Behavior using the chi-square difference test The model chi-square from the full model would be subtracted from the chi-square from a model estimated without the path from gender to Acceptance of Risky Behavior This smaller model has one more degree of freedom and is nested within the larger model If the chi-square difference test is.

since we have chosen W according to a uniform distribution as speci ed in (763) 3 Throw away the worst half of the codewords in the best codebook (n) C Since the arithmetic average probability of error Pe (C ) for this code is less then 2 , we have 1 Pr(E|C ) nR (781) i (C ) 2 , 2 which implies that at least half the indices i and their associated codewords X n (i) must have conditional probability of error i less than 4 (otherwise, these codewords themselves would contribute more than 2 to the sum) Hence the best half of the codewords have a maximal probability of error less than 4 If we reindex these codewords, we have 2nR 1 codewords Throwing out half the code1 words has changed the rate from R to R n , which is negligible for large n.

Building Depreciation Calculation: Purchased a building for $150,000 that has a 25-year life and a residual value of $30,000. (Cost Residual / Life) Depreciation expense: (Cost Residual / Life) Depreciation expense: $150,000 $30,000 / 300 months $400 per month (Cost Residual / Life) Depreciation expense: $150,000 30,000 / 25 years $4,800 per year The adjusting journal entry to recognize depreciation expense for the month of May on the equipment and the building at May 31 for Texana Restaurant is shown in Exhibit 1.9, followed by its posting to the ledger accounts. Depreciation Expense (Expense) Debit Credit Balance $525 $525 Accumulated Depr: Equip. (Contra) Debit Credit Balance $125 $125 Accumulated Depr: Bldg. (Contra) Debit Credit Balance $400 $400

Combining all these improvements, we have constructed a code of rate 1 R = R n , with maximal probability of error (n) 4 This proves the achievability of any rate below capacity Random coding is the method of proof for Theorem 771, not the method of signaling Codes are selected at random in the proof merely to symmetrize the mathematics and to show the existence of a good deterministic code We proved that the average over all codes of block length n has a small probability of error We can nd the best code within this set by an exhaustive search Incidentally, this shows that the Kolmogorov complexity ( 14) of the best code is a small constant This means that the revelation (in step 2) to the sender and receiver of the best code C requires no channel.

The sender and receiver merely agree to use the best (2nR , n) code for the channel Although the theorem shows that there exist good codes with arbitrarily small probability of error for long block lengths, it does not provide a way of constructing the best codes If we used the scheme suggested.