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64d) 67h Two other tests are available for testing the coefficients in a multiple linear regression model: the aligned rank test, proposed by Koul (1970) and developed by Sen and Puri (1977) and Adichie (1978), and the Wald test, proposed by Hettmansperger and McKean (1983) The aligned rank test can be described in terms of the function g(b) in (67) Recall that the estimation procedure in Section 66 is based on the fact that the minimum of g(b) should occur for a vector that is near the parameter vector f3 This implies that the partial derivatives of g(b) should be close to 0 at b = f3; that is, (iJg/iJb)(f3) "" O To test the hypothesis f3 q + I = .. = f3 p = 0, we first estimate f3 under the assumption that the hypothesis is true, by applying the procedure in Section 6.

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1 #include < stream> 2 #include <iostream> 3 using namespace std; 4 5 / / Copy from inFile to outFile 6 void copy( const string & inFile, const string & outFile ) 7 I 8 / / Cheap check for aliasing 9 if( inFile == outFile ) 10 { 11 cerr << "Input and output files are identical\n"; 12 return; 13 } 14 15 / / Open lnput stream 16 ifstream instream( inFi1ec-str( ) ) ; 17 i f ( !instream ) 18 { 19 cerr < < "Can'topen " << inFile << endl; 20 return; 21 I 22 23 / i Open output stream 24 ofstream outstream( outFi1ec-str( ) ) ; 25 lf ( !outstream ) 26 i 27 cerr << "Can'topen " < < outFile < < endl; 28 return; 29 I 30 i / Do the copy 31 char ch; 32 while( inStreamget( ch ) ) 33 if( !outStreamput( ch ) ) 34 { 35 cerr << "Output error!" << endl; 36 return; 37 I 38 1

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able to chain the extraction operators into one statement However, this technique does not work on a few compilers, so instead we use separate extractions After we have done the extractions, we test the stream state We do not need to reset the stream state because each iteration of the w h i l e loop generates a new string stream To avoid the possibility of extraneous inputs, however, we might want to verify that nothing else is left in the string stream The operations that can be performed on string streams are

6 to the reduced model to obtain ~I' .. ' ~q and setting ~q+ I = .. = ~p = O Let Po denote this vector of estimates If the hypothesis is true, then the vector S2 with entries (iJg/iJb)(Po), j = q + 1, .. , p, should be near O The aligned rank test statistic is A = S~V-IS2' where V = X;2Xc2 - X;2Xci(X;IXCI)-IX;lXc2 and Xci and Xc2 are, respectively, columns 1 through q and columns q + 1 through p of the matrix Xc of centered explanatory variables The hypothesis is rejected if A is too large The p-value is calculated from the chi-squared distribution with p - q degrees of freedom See Hettmansperger (1984, Section 532) The phrase "aligned rank" comes from the fact that the test involves the ranks of the residuals Yi - p;)x" which can be regarded as alignments of the observations Yi.

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An aligned rank test uses ranks of residuals, whereas a "pure" rank test uses the ranks of the observations The Wald test of the hypothesis f3 q + I = .. = f3 p = 0 is based directly on the vector of estimates ~2 = (~q+I' .. '~p) from the full model If the hypothesis is true, then P2 should be near O The Wald test statistic is W = P~VP2/( p - q )1'2, where V is from the preceding paragraph and l' is obtained from (69) The hypothesis is rejected if W is too large The p-value is calculated from the F distribution with p - q and n - p - 1 degrees of freedom See Hettmansperger (1984, Section 533) All three tests are equivalent for infinitely large samples In simulation studies with finite samples, Hettmansperger and McKean (1983) found the test in Section 6.

identical to those that can be performed on general streams We use ostringstream objects t o compose character arrays; but using an ostringstream is a bit more complicated, so it is used less often

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