ABSTRACT
It is important to be able to judge the sensitivity of the χ 2 test in distinguishing the null hypothesis from nearby alternatives. We would like to find the probabilities of rejecting the null hypothesis when some relevant alternatives are true; that is, we would like to find the power function of the test. In this section, we find an asymptotic approximation to the power function based on the noncentral χ 2 distribution. In addition to allowing us to measure the sensitivity of the test, this approximation helps solve the important problem of finding the sample size required to obtain a fixed power at a fixed alternative for a given level of significance. The Fix Tables (Table 3) for noncentral χ 2 are in a convenient form for solving this problem for levels 0.05 and 0.01. Fix Tables of Noncentral <italic>χ</italic> <sup>2</sup>. The quantity tabled is that value of the parameter <italic>λ</italic> that satisfies the equation <inline-formula> <alternatives> <mml:math display="inline" xmlns:mml="<a href="https://www.w3.org/1998/Math/MathML" target="_blank">https://www.w3.org/1998/Math/MathML</a>"> <mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mtext>λ</mml:mtext> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:mstyle displaystyle="true"> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi>∞</mml:mi> </mml:munderover> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>λ</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:mrow> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>!</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:mi>Γ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfrac> <mml:mstyle displaystyle="true"> <mml:mrow> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msubsup> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msup> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi>β</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:mrow> </mml:mstyle> </mml:mrow> </mml:mstyle> </mml:mrow> </mml:math> <inline-graphic xlink:href="<a href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136288/c12c5089-77f8-4e14-84a1-364a5e26ae30/content/eq513.tif" target="_blank">https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136288/c12c5089-77f8-4e14-84a1-364a5e26ae30/content/eq513.tif</a>" xmlns:xlink="<a href="https://www.w3.org/1999/xlink" target="_blank">https://www.w3.org/1999/xlink</a>"/> </alternatives> </inline-formula> where <italic>f</italic> = number of degrees of freedom and <italic>χ</italic> <sub> <italic>f</italic> </sub>(<italic>α</italic>) is such that <inline-formula> <alternatives> <mml:math display="inline" xmlns:mml="<a href="https://www.w3.org/1998/Math/MathML" target="_blank">https://www.w3.org/1998/Math/MathML</a>"> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mi>f</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>Γ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfrac> <mml:mstyle displaystyle="true"> <mml:mrow> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msubsup> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msup> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi>α</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:mrow> </mml:mstyle> </mml:mrow> </mml:math> <inline-graphic xlink:href="<a href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136288/c12c5089-77f8-4e14-84a1-364a5e26ae30/content/eq514.tif" target="_blank">https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136288/c12c5089-77f8-4e14-84a1-364a5e26ae30/content/eq514.tif</a>" xmlns:xlink="<a href="https://www.w3.org/1999/xlink" target="_blank">https://www.w3.org/1999/xlink</a>"/> </alternatives> </inline-formula> https://www.niso.org/standards/z39-96/ns/oasis-exchange/table">
α = 0.05
f β
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.426
1.242
2.058
2.911
3.841
4.899
6.172
7.849
10.509
2
0.624
1.731
2.776
3.832
4.957
6.213
7.702
9.635
12.655
3
0.779
2.096
3.302
4.501
5.761
7.154
8.792
10.903
14.172
4
0.910
2.401
3.737
5.050
6.420
7.924
9.683
11.935
15.405
5
1.026
2.667
4.117
5.529
6.991
8.591
10.453
12.828
16.470
6
1.131
2.907
4.458
5.957
7.503
9.187
11.141
13.624
17.419
7
1.228
3.128
4.770
6.349
7.971
9.732
11.768
14.350
18.284
8
1.319
3.333
5.059
6.713
8.405
10.236
12.349
15.022
19.083
9
1.404
3.525
5.331
7.053
8.811
10.708
12.892
15.650
19.829
10
1.485
3.707
5.588
7.375
9.194
11.153
13.404
16.241
20.532
11
1.562
3.880
5.831
7.680
9.557
11.575
13.890
16.802
21.198
12
1.636
4.045
6.064
7.971
9.903
11.977
14.353
17.336
21.833
13
1.707
4.204
6.287
8.250
10.235
12.362
14.796
17.847
22.440
14
1.775
4.357
6.502
8.519
10.554
12.733
15.221
18.338
23.022
15
1.840
4.501
6.709
8.777
10.862
13.090
15.631
18.811
23.583
16
1.904
4.646
6.909
9.027
11.159
13.435
16.027
19.268
24.125
17
1.966
4.784
7.103
9.269
11.447
13.768
16.411
19.710
24.650
18
2.026
4.918
7.291
9.505
11.726
14.092
16.783
20.139
25.158
19
2.085
5.049
7.474
9.734
11.998
14.407
17.144
20.556
25.652
20
2.142
5.176
7.653
9.956
12.262
14.714
17.496
20.961
26.132
α = 0.01
f β
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.674
3.007
4.208
5.394
6.635
8.004
9.611
11.680
14.879
2
2.299
3.941
5.372
6.758
8.190
9.752
11.567
13.881
17.427
3
2.763
4.624
6.218
7.745
9.311
11.008
12.970
15.458
19.248
4
3.149
5.188
6.914
8.557
10.231
12.039
14.121
16.749
20.737
5
3.488
5.682
7.523
9.265
11.033
12.936
15.120
17.871
22.033
6
3.794
6.126
8.069
9.899
11.751
13.738
16.014
18.873
23.187
7
4.075
6.534
8.569
10.480
12.408
14.473
16.831
19.788
24.238
8
4.337
6.912
9.033
11.019
13.017
15.153
17.589
20.636
25.211
9
4.583
7.267
9.469
11.524
13.588
15.790
18.297
21.429
26.122
10
4.816
7.603
9.880
12.000
14.126
16.391
18.965
22.177
26.981
11
5.038
7.922
10.271
12.453
14.638
16.961
19.599
22.887
27.797
12
5.250
8.227
10.644
12.885
15.126
17.505
20.204
23.563
28.575
13
5.453
8.520
11.002
13.299
15.594
18.027
20.784
24.211
29.319
14
5.649
8.801
11.346
13.698
16.043
18.528
21.341
24.833
30.034
15
5.838
9.072
11.678
14.082
16.476
19.011
21.878
25.433
30.722
16
6.021
9.335
11.999
14.454
16.895
19.478
22.396
26.013
31.387
17
6.198
9.590
12.310
14.814
17.301
19.930
22.898
26.574
32.031
18
6.371
9.837
12.612
15.163
17.695
20.369
23.385
27.118
32.655
19
6.539
10.078
12.906
15.502
18.078
20.796
23.859
27.647
33.262
20
6.702
10.312
13.192
15.833
18.451
21.211
24.320
28.162
33.852
