Data Example from Fay, M.P. and Graubard, B.I. "Small Sample Adjustments for Wald-type Tests using Sandwich Esimators" For description of the data see Fay, MP, Freedman, LS, Clifford, CK, Midthune, DN (1997) "Effect of different types and amounts of fat on the development of mammary tumors in rodents: a review" Cancer Research, vol. 57, 3979-3988. OBS=observation number Article = article number set = set (experiment) number n = number of rodents in risk set m = number of rodents that develop mammary tumors restrict = percent calorie restriction pn3 = percent of dietary fat from n-3 PUFA pn6 = percent of dietary fat from n-6 PUFA pzero = percent of dietary fat from saturated fats pmono = percent of dietary fat from monounsaturated fats OBS ARTICLE SET N RESTRICT PN3 PN6 PZERO PMONO M 1 1 205 30 0.0 0.072000000 3.81600000 18.2160000 13.8960000 13 2 1 205 30 0.0 0.108000000 7.66800000 15.3360000 12.8520000 15 3 1 205 30 0.0 0.180000000 12.27600000 11.9160000 11.6640000 16 4 1 205 30 0.0 0.252000000 21.85200000 4.7880000 9.1080000 23 5 1 206 30 0.0 0.000000000 0.90000000 32.7960000 2.3040000 6 6 1 206 30 0.0 0.036000000 2.16000000 31.1040000 2.7000000 10 7 1 206 30 0.0 0.036000000 3.24000000 29.6640000 3.0600000 11 8 1 206 30 0.0 0.036000000 4.06800000 28.5480000 3.3120000 12 9 1 206 30 0.0 0.072000000 6.62400000 25.2000000 4.1400000 15 10 1 206 30 0.0 0.108000000 8.31600000 22.8960000 4.7160000 18 11 1 206 30 0.0 0.180000000 15.91200000 12.7080000 7.2000000 15 12 1 206 30 0.0 0.252000000 21.63600000 5.1120000 9.0360000 21 13 2 23 16 0.0 0.074117647 6.42705882 1.4082353 2.6788235 14 14 2 23 16 0.0 0.367964602 31.90778761 6.9913274 13.2992920 16 15 2 94 27 0.0 0.074117647 6.42705882 1.4082353 2.6788235 9 16 2 94 25 0.0 0.367964602 31.90778761 6.9913274 13.2992920 20 17 2 95 29 0.0 0.074117647 6.42705882 1.4082353 2.6788235 9 18 2 95 29 0.0 0.367964602 31.90778761 6.9913274 13.2992920 17 19 3 1 30 0.0 0.007826087 0.67863354 0.1486957 0.2828571 22 20 3 1 30 0.0 0.252000000 21.85200000 4.7880000 9.1080000 29 21 3 96 28 0.0 0.007826087 0.67863354 0.1486957 0.2828571 4 22 3 96 30 0.0 0.252000000 21.85200000 4.7880000 9.1080000 8 23 3 97 30 0.0 0.007826087 0.67863354 0.1486957 0.2828571 13 24 3 97 30 0.0 0.252000000 21.85200000 4.7880000 9.1080000 23 25 4 19 14 0.0 0.074117647 6.42705882 1.4082353 2.6788235 6 26 4 19 15 0.0 0.343636364 29.79818182 6.5290909 12.4200000 11 27 4 19 14 16.0 0.343636364 29.79818182 6.5290909 12.4200000 1 28 6 2 20 0.0 0.074117647 6.42705882 1.4082353 2.6788235 13 29 6 2 20 0.0 0.198947368 17.25157895 3.7800000 7.1905263 17 30 6 2 20 0.0 0.252000000 21.85200000 4.7880000 9.1080000 16 31 6 2 20 25.0 0.252000000 21.85200000 4.7880000 9.1080000 12 32 6 2 20 25.0 0.314409006 27.26375235 5.9737711 11.3636398 6 33 6 104 20 0.0 0.060000000 5.20285714 1.1400000 2.1685714 16 34 6 104 20 40.0 0.177293233 15.37385607 3.3685714 6.4078840 4 35 7 24 30 0.0 0.074117647 6.42705882 1.4082353 2.6788235 3 36 7 24 30 0.0 0.146187845 12.67657459 2.7775691 5.2836464 2 37 7 24 30 0.0 0.216521739 18.77552795 4.1139130 7.8257143 10 38 7 24 30 0.0 0.286086957 24.80782609 5.4356522 10.3400000 7 39 8 3 20 0.0 0.045542169 3.94915663 0.8653012 1.6460241 9 40 8 3 20 0.0 0.140000000 12.14000000 2.6600000 5.0600000 9 41 8 3 20 0.0 0.252000000 21.85200000 4.7880000 9.1080000 11 42 8 3 20 0.0 2.290120482 0.25373494 1.9908434 1.9778313 7 43 8 3 20 0.0 7.040000000 0.78000000 6.1200000 6.0800000 4 44 8 3 20 0.0 12.672000000 1.40400000 11.0160000 10.9440000 6 45 8 44 20 0.0 0.000000000 4.46963855 0.7091566 1.3272289 14 46 8 44 20 0.0 0.000000000 13.74000000 2.1800000 4.0800000 16 47 8 44 20 0.0 0.000000000 24.73200000 3.9240000 7.3440000 18 48 8 44 20 0.0 0.000000000 4.40000000 13.5400000 2.0800000 17 49 8 44 20 0.0 0.000000000 4.28400000 28.6920000 2.9880000 15 50 9 4 24 0.0 0.252000000 21.85200000 4.7880000 9.1080000 18 51 9 4 26 0.0 0.007826087 0.67863354 0.1486957 0.2828571 5 52 10 34 20 0.0 0.105882353 2.19176471 3.7482353 4.5423529 17 53 10 34 20 0.0 0.422009569 5.57052632 16.7115789 19.4968421 19 54 10 35 20 0.0 0.105882353 2.19176471 3.7482353 4.5423529 16 55 10 35 20 0.0 0.435071090 5.74293839 17.2288152 20.1002844 14 56 10 36 20 0.0 0.105882353 2.19176471 3.7482353 4.5423529 16 57 10 36 20 0.0 0.435071090 5.74293839 17.2288152 20.1002844 19 58 10 37 20 0.0 0.105882353 2.19176471 3.7482353 4.5423529 17 59 10 37 20 0.0 0.435071090 5.74293839 17.2288152 20.1002844 16 60 11 101 20 0.0 0.074117647 6.42705882 1.4082353 2.6788235 12 61 11 101 20 10.0 0.082429907 7.14785047 1.5661682 2.9792523 12 62 11 101 20 20.0 0.091981460 7.97610660 1.7476477 3.3244728 8 63 11 101 20 30.0 0.102709529 8.90638347 1.9514811 3.7122158 7 64 11 101 20 40.0 0.118437146 10.27019253 2.2503058 4.2806569 1 65 15 45 20 0.0 0.000000000 0.68400000 33.1200000 2.2320000 15 66 15 45 20 0.0 0.000000000 24.73200000 3.9240000 7.3440000 19 67 15 45 20 0.0 0.000000000 1.12554217 4.7624096 0.6115663 14 68 15 45 20 0.0 0.000000000 0.57600000 28.1160000 7.2720000 15 69 15 45 20 0.0 0.000000000 5.97600000 28.1160000 1.8720000 20 70 15 45 20 0.0 1.944000000 0.79200000 29.7720000 3.5280000 18 71 16 5 24 0.0 0.067100592 5.81857988 1.2749112 2.4252071 18 72 16 5 23 0.0 0.252000000 21.85200000 4.7880000 9.1080000 18 73 17 25 30 0.0 0.074117647 6.42705882 1.4082353 2.6788235 20 74 17 25 30 0.0 0.286086957 24.80782609 5.4356522 10.3400000 26 75 17 25 30 0.0 0.042352941 8.24823529 0.9423529 1.3447059 19 76 17 25 30 0.0 0.163478261 31.83739130 3.6373913 5.1904348 26 77 17 25 30 0.0 0.063529412 0.87882353 1.4611765 8.1741176 18 78 17 25 30 0.0 0.245217391 3.39217391 5.6400000 31.5513043 19 79 17 25 30 0.0 0.000000000 0.77652174 37.6000000 2.5339130 14 80 18 6 25 0.0 0.074117647 6.42705882 1.4082353 2.6788235 11 81 18 6 25 0.0 0.252000000 21.85200000 4.7880000 9.1080000 20 82 18 6 25 0.0 0.042352941 0.78352941 3.7694118 5.9929412 4 83 18 6 25 0.0 0.144000000 2.66400000 12.8160000 20.3760000 8 84 18 6 25 0.0 0.042352941 0.78352941 3.7694118 5.9929412 6 85 18 6 25 0.0 0.144000000 2.66400000 12.8160000 20.3760000 10 86 19 73 23 0.0 0.074117647 6.42705882 1.4082353 2.6788235 9 87 19 73 23 0.0 0.370326567 32.11260371 7.0362048 13.3846602 18 88 19 76 30 0.0 0.074117647 6.42705882 1.4082353 2.6788235 11 89 19 76 30 0.0 0.370326567 32.11260371 7.0362048 13.3846602 24 90 19 77 30 0.0 0.074117647 6.42705882 1.4082353 2.6788235 6 91 19 77 30 0.0 0.370326567 32.11260371 7.0362048 13.3846602 17 92 19 78 29 0.0 0.074117647 6.42705882 1.4082353 2.6788235 1 93 19 78 30 0.0 0.370326567 32.11260371 7.0362048 13.3846602 11 94 19 79 30 0.0 0.074117647 6.42705882 1.4082353 2.6788235 0 95 19 79 30 0.0 0.370326567 32.11260371 7.0362048 13.3846602 8 96 22 106 11 0.0 1.851428571 7.14122449 8.2983673 15.8032653 11 97 22 106 10 46.0 2.792903226 3.87483871 5.8877419 12.5051613 1 98 22 107 21 0.0 1.851428571 7.14122449 8.2983673 15.8032653 21 99 22 107 12 46.0 2.792903226 3.87483871 5.8877419 12.5051613 2 100 23 26 23 0.0 0.074117647 6.42705882 1.4082353 2.6788235 8 101 23 26 26 0.0 0.300000000 26.01428571 5.7000000 10.8428571 22 102 23 26 27 0.0 0.428571429 5.22857143 17.1428571 19.9714286 17 103 23 26 30 0.0 0.300000000 2.18571429 21.9000000 18.5142857 15 104 23 26 23 0.0 0.000000000 1.58571429 38.3571429 2.8714286 10 105 24 7 25 0.0 0.074117647 6.42705882 1.4082353 2.6788235 15 106 24 7 24 0.0 0.252000000 21.85200000 4.7880000 9.1080000 16 107 26 38 20 0.0 0.000000000 4.46963855 0.7091566 1.3272289 14 108 26 38 20 0.0 0.216000000 1.18800000 18.8280000 15.7320000 14 109 26 38 20 0.0 0.000000000 0.68400000 33.1200000 2.2320000 15 110 26 38 20 0.0 0.396000000 3.85200000 14.7600000 17.0280000 18 111 26 38 20 0.0 0.000000000 24.73200000 3.9240000 7.3440000 17 112 26 38 20 0.0 0.216000000 4.75200000 16.5960000 14.5080000 18 113 26 38 20 0.0 0.000000000 4.28400000 28.6920000 2.9880000 17 114 29 8 21 0.0 0.007826087 0.67863354 0.1486957 0.2828571 15 115 29 8 22 0.0 0.252000000 21.85200000 4.7880000 9.1080000 21 116 29 8 21 0.0 0.000000000 0.68400000 33.1200000 2.2320000 16 117 31 43 32 0.0 0.074117647 6.42705882 1.4082353 2.6788235 15 118 31 43 32 0.0 0.252000000 21.85200000 4.7880000 9.1080000 18 119 31 43 32 0.0 0.072000000 3.45600000 18.5040000 13.9680000 16 120 31 43 32 0.0 0.396000000 3.85200000 14.7600000 17.0280000 25 121 31 43 32 0.0 0.216000000 1.18800000 18.8280000 15.7320000 21 122 33 46 9 0.0 0.393540373 0.04360248 0.3421118 0.3398758 9 123 33 46 10 0.0 2.290120482 0.25373494 1.9908434 1.9778313 10 124 33 46 8 0.0 12.672000000 1.40400000 11.0160000 10.9440000 5 125 33 47 11 0.0 0.393540373 0.04360248 0.3421118 0.3398758 11 126 33 47 10 0.0 0.007826087 0.67863354 0.1486957 0.2828571 10 127 33 47 8 0.0 12.672000000 1.40400000 11.0160000 10.9440000 5 128 33 47 9 0.0 0.252000000 21.85200000 4.7880000 9.1080000 9 129 34 105 17 0.0 0.074117647 6.42705882 1.4082353 2.6788235 12 130 34 105 14 37.0 0.074117647 6.42705882 1.4082353 2.6788235 0 131 36 87 30 0.0 0.062123894 3.35469027 9.4221239 7.8483186 20 132 36 87 30 0.0 0.088117155 9.83974895 9.7222594 9.7516318 23 133 36 87 30 0.0 0.088117155 4.75832636 13.3644351 11.1321339 23 134 36 87 30 0.0 0.115221239 6.22194690 17.4752212 14.5562832 23 135 36 87 30 30.0 0.062123894 3.35469027 9.4221239 7.8483186 15 136 36 87 30 30.0 0.088117155 9.83974895 9.7222594 9.7516318 17 137 36 87 30 30.0 0.088117155 4.75832636 13.3644351 11.1321339 19 138 36 87 30 30.0 0.115221239 6.22194690 17.4752212 14.5562832 17 139 38 33 19 0.0 0.045542169 3.94915663 0.8653012 1.6460241 15 140 38 33 19 0.0 0.252000000 21.85200000 4.7880000 9.1080000 15 141 42 103 70 20.0 0.074117647 6.42705882 1.4082353 2.6788235 41 142 42 103 70 50.0 0.074117647 6.42705882 1.4082353 2.6788235 8 143 43 9 25 0.0 0.074117647 6.42705882 1.4082353 2.6788235 12 144 43 9 25 0.0 0.252000000 21.85200000 4.7880000 9.1080000 21 145 43 10 25 0.0 0.074117647 6.42705882 1.4082353 2.6788235 10 146 43 10 25 0.0 0.252000000 21.85200000 4.7880000 9.1080000 19 147 44 112 48 0.0 0.396000000 3.85200000 14.7600000 17.0280000 32 148 44 112 44 30.0 0.317880378 3.09210913 11.8482686 13.6688562 0 149 45 11 30 0.0 0.007826087 0.67863354 0.1486957 0.2828571 21 150 45 11 30 0.0 0.074117647 6.42705882 1.4082353 2.6788235 23 151 45 11 30 0.0 0.140000000 12.14000000 2.6600000 5.0600000 28 152 45 11 30 0.0 0.252000000 21.85200000 4.7880000 9.1080000 27 153 45 11 30 0.0 0.000000000 0.68400000 33.1200000 2.2320000 29 154 45 11 30 0.0 0.216000000 0.68400000 22.6080000 12.4920000 26 155 45 11 30 0.0 0.216000000 1.18800000 18.8280000 15.7320000 24 156 45 11 30 0.0 0.396000000 3.85200000 14.7600000 17.0280000 28 157 45 11 30 0.0 0.216000000 2.98800000 4.9680000 27.7920000 26 158 45 11 30 0.0 2.916000000 5.83200000 1.5480000 25.6680000 23 159 45 11 30 0.0 0.000000000 21.27600000 8.0640000 6.6600000 28 160 45 11 30 0.0 2.556000000 19.29600000 5.3640000 8.7480000 30 161 45 11 30 0.0 0.000000000 24.73200000 3.9240000 7.3440000 29 162 46 12 22 0.0 0.007826087 0.67863354 0.1486957 0.2828571 3 163 46 12 22 0.0 0.252000000 21.85200000 4.7880000 9.1080000 15 164 46 208 22 0.0 0.007826087 0.67863354 0.1486957 0.2828571 9 165 46 208 22 0.0 0.252000000 21.85200000 4.7880000 9.1080000 22 166 47 48 30 0.0 0.074117647 6.42705882 1.4082353 2.6788235 18 167 47 48 30 0.0 0.281359223 24.39786408 5.3458252 10.1691262 23 168 47 48 30 0.0 0.081739130 6.21217391 32.0417391 2.5747826 17 169 49 90 12 0.0 0.338150289 3.40855491 3.4897110 6.2760694 9 170 49 90 12 0.0 1.182857143 16.75714286 17.1514286 30.5571429 11 171 49 90 12 15.0 1.182857143 16.75714286 17.1514286 30.5571429 12 172 49 90 10 37.0 1.182857143 16.75714286 17.1514286 30.5571429 5 173 49 90 9 32.0 0.478681319 5.52659341 5.6571429 10.1175824 9 174 49 90 12 26.0 0.338150289 3.40855491 3.4897110 6.2760694 12 175 50 16 49 0.0 0.074117647 6.42705882 1.4082353 2.6788235 35 176 50 16 28 0.0 0.296328872 25.69594646 5.6302486 10.7101721 23 177 52 15 30 0.0 0.007826087 0.67863354 0.1486957 0.2828571 4 178 52 15 28 0.0 0.074117647 6.42705882 1.4082353 2.6788235 13 179 52 15 28 0.0 0.252000000 21.85200000 4.7880000 9.1080000 21 180 53 14 17 0.0 0.067100592 5.81857988 1.2749112 2.4252071 15 181 53 14 18 0.0 0.252000000 21.85200000 4.7880000 9.1080000 18 182 54 20 40 0.0 0.074117647 6.42705882 1.4082353 2.6788235 15 183 54 20 40 0.0 0.152307692 13.20725275 2.8938462 5.5048352 20 184 54 20 40 0.0 0.300000000 26.01428571 5.7000000 10.8428571 30 185 54 21 40 0.0 0.074117647 6.42705882 1.4082353 2.6788235 15 186 54 21 40 0.0 0.152307692 13.20725275 2.8938462 5.5048352 14 187 54 21 40 0.0 0.300000000 26.01428571 5.7000000 10.8428571 26 188 54 22 40 0.0 0.074117647 6.42705882 1.4082353 2.6788235 12 189 54 22 40 0.0 0.152307692 13.20725275 2.8938462 5.5048352 24 190 54 22 40 0.0 0.300000000 26.01428571 5.7000000 10.8428571 28 191 55 13 34 0.0 0.074117647 6.42705882 1.4082353 2.6788235 15 192 55 13 32 0.0 0.231428571 20.06816327 4.3971429 8.3644898 19 193 56 39 20 0.0 0.116470588 1.13294118 4.3411765 5.0082353 15 194 56 39 20 0.0 0.396000000 3.85200000 14.7600000 17.0280000 19 195 56 40 20 0.0 0.116470588 1.13294118 4.3411765 5.0082353 14 196 56 40 20 0.0 0.396000000 3.85200000 14.7600000 17.0280000 19 197 56 352 20 0.0 0.116470588 1.13294118 4.3411765 5.0082353 8 198 56 352 20 0.0 0.396000000 3.85200000 14.7600000 17.0280000 19 199 56 353 20 0.0 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