Appendix: Statistical Tables

Appendix: Statistical Tables

Table 1 Standard normal probabilities (area between 0 and z)

779B. Lawal, Applied Statistical Methods in Agriculture,Health and Life Sciences, DOI 10.1007/978-3-319-05555-8,c© Springer International Publishing Switzerland 2014

780 Appendix: Statistical Tables

Table 2 Values of tα in a t distribution with df degrees of freedom. (shaded areaP (t > tα) = α)

Appendix: Statistical Tables 781

Table 3 Values of χ2α,df in a chi-square distribution with df degrees of freedom

(shaded area P (χ2 > χ2α,df) = α)


Table 4 Values of fα,ν1,ν2 in an F distribution (shaded area P (F > fα,ν1,ν2) = α).Numerator degrees of freedom is ν1 and denominator degrees of freedom is ν2.


Table 4 Values of fα,ν1,ν2 in an F distribution (continued)


Table 5 Orthogonal polynomial coefficients

Coefficients

k Polynomial 1 2 3 4 5 6 7 8∑

c2i

3 Linear −1 0 1 2Quadratic 1 −2 1 6

4 Linear −3 −1 1 3 20Quadratic 1 −1 −1 1 4Cubic −1 3 −3 1 20

5 Linear −2 −1 0 1 2 10Quadratic 2 −1 −2 −1 2 14Cubic −1 2 0 −2 1 10Quartic 1 −4 6 −4 1 70

6 Linear −5 −3 −1 1 3 5 70Quadratic 5 −1 −4 −4 −1 5 84Cubic −5 7 4 −4 −7 5 180Quartic 1 −3 2 2 −3 1 28Quintic −1 5 −10 10 −5 1 252

7 Linear −3 −2 −1 0 1 2 3 28Quadratic 5 0 −3 −4 −3 0 5 84Cubic −1 1 1 0 −1 −1 1 6Quartic 3 −7 1 6 1 −7 3 154Quintic −1 4 −5 0 5 −4 1 84Sextic 1 −6 15 −20 15 −6 1 924

8 Linear −7 −5 −3 −1 1 3 5 7 168Quadratic 7 1 −3 −5 −5 −3 1 7 168Cubic −7 5 7 3 −3 −7 −5 7 264Quartic 7 −13 −3 9 9 −3 −13 7 616Quintic −7 23 −17 −15 15 17 −23 7 2184Sextic 1 −5 9 −5 −5 9 −5 1 264Septic −1 7 −21 35 −35 21 −7 1 3432


Table 6 Upper α point of Studentized range, qα(k, ν), where k = r = number oftreatments to be compared and ν = the number of degrees of freedom. (Source: TheAnalysis of Variance by Scheffe, H. (1959). Reproduced with permission from JohnWiley & Sons, Inc.)

Table 7 Upper α point of Studentized range (continued)


Table 8 Upper α = 0.05 points of Duncan’s multiple range tests

k

df 2 3 4 5 6 7 8 9 10

1 17.969 17.969 17.969 17.969 17.969 17.969 17.969 17.969 17.9692 6.085 6.085 6.085 6.085 6.085 6.085 6.085 6.085 6.0853 4.501 4.516 4.516 4.516 4.516 4.516 4.516 4.516 4.5164 3.926 4.013 4.033 4.033 4.033 4.033 4.033 4.033 4.0335 3.635 3.749 3.796 3.814 3.814 3.814 3.814 3.814 3.8146 3.460 3.586 3.649 3.680 3.694 3.697 3.697 3.697 3.6977 3.344 3.477 3.548 3.588 3.611 3.622 3.625 3.625 3.6258 3.261 3.398 3.475 3.521 3.549 3.566 3.575 3.579 3.5799 3.199 3.339 3.420 3.470 3.502 3.523 3.536 3.544 3.54710 3.151 3.293 3.376 3.430 3.465 3.489 3.505 3.516 3.52211 3.113 3.256 3.341 3.397 3.435 3.462 3.480 3.493 3.50112 3.081 3.225 3.312 3.370 3.410 3.439 3.459 3.474 3.48413 3.055 3.200 3.288 3.348 3.389 3.419 3.441 3.458 3.47014 3.033 3.178 3.268 3.328 3.371 3.403 3.426 3.444 3.45715 3.014 3.160 3.250 3.312 3.356 3.389 3.413 3.432 3.44616 2.998 3.144 3.235 3.297 3.343 3.376 3.402 3.422 3.43717 2.984 3.130 3.222 3.285 3.331 3.365 3.392 3.412 3.42918 2.971 3.117 3.210 3.274 3.320 3.356 3.383 3.404 3.42119 2.960 3.106 3.199 3.264 3.311 3.347 3.375 3.397 3.41520 2.950 3.097 3.190 3.255 3.303 3.339 3.368 3.390 3.40921 2.941 3.088 3.181 3.247 3.295 3.332 3.361 3.385 3.40322 2.933 3.080 3.173 3.239 3.288 3.326 3.355 3.379 3.39823 2.926 3.072 3.166 3.233 3.282 3.320 3.350 3.374 3.39424 2.919 3.066 3.160 3.226 3.276 3.315 3.345 3.370 3.39025 2.913 3.059 3.154 3.221 3.271 3.310 3.341 3.366 3.38626 2.907 3.054 3.149 3.216 3.266 3.305 3.336 3.362 3.38227 2.902 3.049 3.144 3.211 3.262 3.301 3.332 3.358 3.37928 2.897 3.044 3.139 3.206 3.257 3.297 3.329 3.355 3.37629 2.892 3.039 3.135 3.202 3.253 3.293 3.326 3.352 3.37330 2.888 3.035 3.131 3.199 3.250 3.290 3.322 3.349 3.37131 2.884 3.031 3.127 3.195 3.246 3.287 3.319 3.346 3.36832 2.881 3.028 3.123 3.192 3.243 3.284 3.317 3.344 3.36633 2.877 3.024 3.120 3.188 3.240 3.281 3.314 3.341 3.36434 2.874 3.021 3.117 3.185 3.238 3.279 3.312 3.339 3.36235 2.871 3.018 3.114 3.183 3.235 3.276 3.309 3.337 3.36036 2.868 3.015 3.111 3.180 3.232 3.274 3.307 3.335 3.35837 2.865 3.013 3.109 3.178 3.230 3.272 3.305 3.333 3.35638 2.863 3.010 3.106 3.175 3.228 3.270 3.303 3.331 3.35539 2.861 3.008 3.104 3.173 3.226 3.268 3.301 3.330 3.35340 2.858 3.005 3.102 3.171 3.224 3.266 3.300 3.328 3.35248 2.843 2.991 3.087 3.157 3.211 3.253 3.288 3.318 3.34260 2.829 2.976 3.073 3.143 3.198 3.241 3.277 3.307 3.33380 2.814 2.961 3.059 3.130 3.185 3.229 3.266 3.297 3.323120 2.800 2.947 3.045 3.116 3.172 3.217 3.254 3.286 3.313240 2.786 2.933 3.031 3.103 3.159 3.205 3.243 3.276 3.304Inf 2.772 2.918 3.017 3.089 3.146 3.193 3.232 3.265 3.294


Table 9 Upper α = 0.01 point of Duncan’s multiple range tests

k

df 2 3 4 5 6 7 8 9 10 11

1 90.024 90.024 90.024 90.024 90.024 90.024 90.024 90.024 90.024 90.0242 14.036 14.036 14.036 14.036 14.036 14.036 14.036 14.036 14.036 14.0363 8.260 8.321 8.321 8.321 8.321 8.321 8.321 8.321 8.321 8.3214 6.511 6.677 6.740 6.755 6.755 6.755 6.755 6.755 6.755 6.7555 5.702 5.893 5.989 6.040 6.065 6.074 6.074 6.074 6.074 6.0746 5.243 5.439 5.549 5.614 5.655 5.680 5.694 5.701 5.703 5.7037 4.949 5.145 5.260 5.333 5.383 5.416 5.439 5.454 5.464 5.4708 4.745 4.939 5.056 5.134 5.189 5.227 5.256 5.276 5.291 5.3029 4.596 4.787 4.906 4.986 5.043 5.086 5.117 5.142 5.160 5.17410 4.482 4.671 4.789 4.871 4.931 4.975 5.010 5.036 5.058 5.07411 4.392 4.579 4.697 4.780 4.841 4.887 4.923 4.952 4.975 4.99412 4.320 4.504 4.622 4.705 4.767 4.815 4.852 4.882 4.907 4.92713 4.260 4.442 4.560 4.643 4.706 4.754 4.793 4.824 4.850 4.87114 4.210 4.391 4.508 4.591 4.654 4.703 4.743 4.775 4.802 4.82415 4.167 4.346 4.463 4.547 4.610 4.660 4.700 4.733 4.760 4.78316 4.131 4.308 4.425 4.508 4.572 4.622 4.662 4.696 4.724 4.74817 4.099 4.275 4.391 4.474 4.538 4.589 4.630 4.664 4.692 4.71718 4.071 4.246 4.361 4.445 4.509 4.559 4.601 4.635 4.664 4.68919 4.046 4.220 4.335 4.418 4.483 4.533 4.575 4.610 4.639 4.66420 4.024 4.197 4.312 4.395 4.459 4.510 4.552 4.587 4.617 4.64221 4.004 4.177 4.291 4.374 4.438 4.489 4.531 4.567 4.597 4.62222 3.986 4.158 4.272 4.355 4.419 4.470 4.513 4.548 4.578 4.60423 3.970 4.141 4.254 4.337 4.402 4.453 4.496 4.531 4.562 4.58824 3.955 4.126 4.239 4.322 4.386 4.437 4.480 4.516 4.546 4.57325 3.942 4.112 4.224 4.307 4.371 4.423 4.466 4.502 4.532 4.55926 3.930 4.099 4.211 4.294 4.358 4.410 4.452 4.489 4.520 4.54627 3.918 4.087 4.199 4.282 4.346 4.397 4.440 4.477 4.508 4.53528 3.908 4.076 4.188 4.270 4.334 4.386 4.429 4.465 4.497 4.52429 3.898 4.065 4.177 4.260 4.324 4.376 4.419 4.455 4.486 4.51430 3.889 4.056 4.168 4.250 4.314 4.366 4.409 4.445 4.477 4.50431 3.881 4.047 4.159 4.241 4.305 4.357 4.400 4.436 4.468 4.49532 3.873 4.039 4.150 4.232 4.296 4.348 4.391 4.428 4.459 4.48733 3.865 4.031 4.142 4.224 4.288 4.340 4.383 4.420 4.452 4.47934 3.859 4.024 4.135 4.217 4.281 4.333 4.376 4.413 4.444 4.47235 3.852 4.017 4.128 4.210 4.273 4.325 4.369 4.406 4.437 4.46536 3.846 4.011 4.121 4.203 4.267 4.319 4.362 4.399 4.431 4.45937 3.840 4.005 4.115 4.197 4.260 4.312 4.356 4.393 4.425 4.45238 3.835 3.999 4.109 4.191 4.254 4.306 4.350 4.387 4.419 4.44739 3.830 3.993 4.103 4.185 4.249 4.301 4.344 4.381 4.413 4.44140 3.825 3.988 4.098 4.180 4.243 4.295 4.339 4.376 4.408 4.43648 3.793 3.955 4.064 4.145 4.209 4.261 4.304 4.341 4.374 4.40260 3.762 3.922 4.030 4.111 4.174 4.226 4.270 4.307 4.340 4.36880 3.732 3.890 3.997 4.077 4.140 4.192 4.236 4.273 4.306 4.335120 3.702 3.858 3.964 4.044 4.107 4.158 4.202 4.239 4.272 4.301240 3.672 3.827 3.932 4.011 4.073 4.125 4.168 4.206 4.239 4.268Inf 3.643 3.796 3.900 3.978 4.040 4.091 4.135 4.172 4.205 4.235

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Index

AAdditivity, 397, 425, 427, 429Aliases, 595, 597, 599, 605Alpha level-α, 147, 171, 179, 358Alternative hypothesis, 134, 135, 142,

154, 205, 340Analysis of Variance Table-Oneway

Regression, 226, 245RCBD, 403

ANOVA, see Analysis of VarianceArray, 11, 38, 39, 45Association, 309

BBalanced incomplete block design, 641

ANOVA table, 644examples, 643parameter equations, 641parameters relationship, 641reduced design, 641statistical model, 642

Bartlett’s homogeneity test, 186, 764Bayes’s Theorem, 74Bias in measurement, 8BIBD construction, 642, 652Binomial distribution, 80, 82

fitting, 327–330Box-Cox transformation, 438

CCalculating factorial interaction

contrasts, 557Case-control studies, 423Categorical variable

nominal, 4, 307ordinal, 307

Censoring, 719

interval censoring, 720left censoring, 720right censoring, 720

Charts, 24bar chart, 25component chart, 28multiple bar chart, 26, 27pie chart, 28, 29

Chi squared distribution, 312degree of freedom, 312table, 781

Classinterval, 13limits, 14

Coefficient of Determination, 241,242, 304

Coefficient of Variation (CV), 50, 51, 635Cohort studies, 348Combination, 62Combined analysis of experiments, 749

over several seasons, 750over several sites, 763over several years, 771

Combining several 2 × 2 contin-gency tables, 315–317, see alsoMantel-Haenzel test

Comparisons of Regressions, 257Completely Randomized Design

(CRD), 355correction factor, 357hypotheses of interest, 369pilot studies, 355

Concordance correlation, 253Conditional probability, 67, 727, 735Confidence difference for two

populations, 123for means (μ1 − μ2), 123


794 Index

for proportions (p1 − p2), 127Confidence interval, 115

confidence coefficient, 117difference of two means, 123

Confidence interval estimates for β1, 227for β0, 229

Confidence interval for a proportion, 121Confounding in factorial designs, 572

partial confounding, 582Constant variance, 185, 186, 217, 218,

236, 386, 387, 430, 451Contingency tables

assumptions and rule, 325expected values, 325general r × c table, 319homogeneity model and test, 320the 2 × 2 table, 315

Contrasts, 370contrasts sum of squares, 370orthogonal contrast, 370

Cook’s measure, 249Corrected sum of cross-products

of xy, 218, 219corrected sum of squares

of x, 218, 219corrected sum of squares

of y, 223, 224Correlation coefficient, 241

general hypotheses, 242properties of r, 241sample correlation coefficient

r, 241, 304Covariance analysis, 503

adjusted treatment mean, 507adjusted treatment SS, 507ANOVA table, 518, 519assumptions, 505concomitant variable, 503in Factorial Designs, 520in Randomized Complete Block

Design, 516missing values and Covariance

Analysis, 523parallelism test, 508

Critical rejection region, 135, 136Cross-over designs, 475

carry over effect, 484direct effect, 475residual effect, 475sequential effects, 484

Cumulative probability distribution, 79

DDegrees of freedom (df)

for X2 distribution, 322for F distribution, 171for t distribution, 145, 159, 176,

228, 229Dependent variable, 217, 249, 471,

520, 697Descriptive statistics, 41, 100, 101Discrete random variable, 77

density function, 78mean, 78variance, 78

The dot plot, see Graphicalrepresentation

Drug responsiveness model, 290an example, 290

Duncan’s multiple range test, 361tables, 786, 787

EEfficiency factor in BIB design, 648Empirical rule, 54Error in measurement, 5, 8Estimator, 115

estimate, 115Expected frequency, 308, 317, 326Experiments, 307, 308Experimental units, 338Experiments over years, 771Explanatory variable, 217Exponential

hazard model, 737response model, 284

Mitscherlich fertilizer responsemodel, 284

FF distribution, 147, 171, 249, 265,

358, 360table, 782, 783

Factorial designs, 531in complete blocks, 546other factorial systems, 553the 22 factorial, 532the 2n factorial design, 532treatment combinations, 531

False negative, 70False positive, 70Fisher’s exact test, 312Fixed effects, 347Five number summary, 41, 57Follow-up studies, 348

Index 795

Fractional factorial design, 590aliased effects, 591complimentary fraction, 590, 592principal fraction, 590, 592

Frequencyexpected, 308, 317, 326observed, 307, 308, 317, 319

Frequency distribution, 11construction, 14cumulative frequency, 17grouped distribution, 13relative frequency, 15ungrouped distribution, 12

Frequency polygons, 23, 24ogive, 23, 24

GGeometric mean, 36Goodness-of-fit tests, 325, 327Graeco Latin squares, 461, 462, 466Graphical representation, 18

dotplot, 18histogram, 21stem and leaf display, 18two-stemmer, 19

Group balanced block design, 440ANOVA structure, 442example, 442randomization, 441

HHazard function, 735Histograms, 21

construction, 21Homogeneity of Variances, 185, 186Homoscedasticity, 219Horizontal strip plot, 630Hyper-geometric distribution, 96Hypothesis testing, 128

alternative, 134null, 130, 131

IIncomplete block design, 639

confounded with blocks, 639concurrence λij, 640example, 640

Independent variable, 243, 251, 254, 255,298, 520, 531

Influential observations, 249Cooks’s measure, 249Dffits, 249leverages, 249

Interaction effects, 533, 535

qualitative interaction, 537quantitative interaction, 537

KKaplan-Meier estimators, 720

method, 720

LLack of fit sum of squares, 235Lattice design, 649

ANOVA structure, 652an example, 652construction, 649randomization, 651

Latin square designs, 451analysis, 453ANOVA Table, 454, 457stratification, 453The Completely Orthogonalized

Square, 465Laws of Probability, 66LD50, 661Least Significance Difference (LSD),

176, 360Least squares, 217, 219Level of a factor, 531Level of significance, 133Life-Table method, 724Linear

combinations of means, 369, 370logistic model, 662, 664, 666

Logarithmic transformation, 431, 433Logits, 665Logistic regression, 664Log dose, 664

MMain effects, 532Main plots, 609, 610

assignment of factors, 627randomization, 637

Margin of error, 4, 118, 121, 122, 164,167

Marginal probabilities, 65, 66Measures of center, 34

relationships between themeasures, 46

Multiple linear regression, 242Mean, 35

for grouped data, 36, 42weighted mean, 36

Mean of discrete random variable, 78Median, 37

796 Index

for grouped data, 43Measures of variability, 49

range, 49standard deviation, 50

for grouped data, 52variance, 50

Measurement, 4Michaelis-Menten model, 302MINITAB, 18, 21, 44Missing Values

ANOVA table, 409, 460in Latin Square Designs, 459in RCBD, 407

Mitscherlic Response model, seeMitscherlich fertilizer responsemodel

Mode, 44Model adequacy, 234

model adequacy testing, 386Model assumptions, 218Mortality rates, 664Multicollinearity, 251

Variance Inflation Factor (VIF), 251Multiple comparison tests, 175, 360

Duncan’s multiple range test, 361in RCBD, 422least significance difference, 360Scheffe’s test, 362t pairwise tests, 175t tests, 360Tukey’s test, 362

Multiple Latin Squares, 468Multiple and partial correlations, 254

NNegative exponential model, 271Non-additivity

Tukey’s additivity test, 425Normal approximation to binomial, 94Normality assumption test

Anderson-Darling test, 387Normal distribution

standardized normal, 88Normal probability plot, 236Nonlinear regression, 269Non-parametric tests

Kruskal-Wallis test, 189Mann-Whitney U test, 150

Null hypothesis, 133

OObservational studies, 348Odds of an event, 351

relationship with probability, 69Odds interpretation, 667Odds ratio, 351One sample tests

one sample t test, 138one sample Z test, 135

Ordinary least squares, 218, 219Orthogonal contrasts, 181Orthogonal designs, 348

in Latin Squares Design, 476Orthogonal polynomials

cubic effect coefficients, 288, 379equally Spaced treatment levels,

285, 375linear effect coefficients, 288, 379quadratic effect coefficients, 288, 379table of coefficients, 603, 784unequal spacing, 1, 48, 198

Outliers, 40, 249

PPaired t Test, 154Parallel bio-assay, 674

use of joint model, 676Parameters

estimates, 225interpretations, 225, 688

Parallel regression lines, 268Partial F tests, 247Percentiles, 39

quartiles, 40lower quartile, 40upper quartile, 40

Partitioning treatment SS, 382, 414in RCBD, 412

Pearson’s X2 Statistic, 352Percentage variation, 233Permutation, 60Pie chart, 28Pivot cell in 2 × 2 contingency table, 442Poisson distribution, 85

fitting, 86, 325recursion formula, 87

Polynomial regression, 298Pooled variance estimator, 125Pooled t test, 150Prediction of Y from X

individual response, 231mean response, 231

Predictive value positive, 70, 72, 73Probability, 59

laws of, 66Probability of an event, 63Probability distribution, 77

Index 797

Probability density function (pdf), 78Probability tree, 72Probit analysis, 671

probit versus logistic regression, 673Product-Limit method, 746, 747Proportional hazard model, 737, 744Prospective studies

retrospective studies, 350definition, 248

P-valuefor X2 test, 311, 325for F test, 147for t test, 140for Z test, 135

Pure error SS, 234

QQuadratic model, 242Quantal-bioassay

individual effective dose, 661median Lethal Dose, LD50, 661, 670median Effective Dose, ED50, 661median Lethal Concentration,

LC50, 661median Effective Concentration,

EC50, 661examples, 662

Quartileslower, 39outliers from, 40upper, 39

Quantitative treatment levels, 375Qualitative

factor, 376variable, 24

Quantitativefactor, 338, 376variable, 9

Questionnaires, 6, 7

RRandom error term, 317Randomized Complete Block

Design, 395Analysis of Variance table, 404, 405blocking, 398group balanced block design, see

Group Balanced Block Designmatching, 402model and analysis, 402

Rank correlation, 252Raw data, 11Random variable, 77Regression

analysis, 217assumptions, 217, 218

Relative efficiencyin Balanced Lattice Design, 657in BIB design, 648in RCBD, 467in Latin Square Design, 459

Relative risk, 349ROC curve, 74–77Repeated measures design

within-subject method, 697Replicated factorial design, 538Replication

definition of, 342determination of number, 202

Residuals, 230examination of, 236

Response, 217Resolution III designs, 592Resolution IV designs, 596Retrospective studies, 320, 348, 350

SSampling distributions, 98

of x, 105Central Limit theorem, 104

summary results, 79of proportion p, 106Sample size determination, 118, 202

in proportion, 121Sample space, 59, 62Scatter plot, 273Sensitivity, see SpecificityShape

bell shaped, 54, 88left skewed, 47mound shaped, 54right skewed, 47

Significance levels, 132, 133Simple events, 62Simple factorial effects, 582Simpson’s paradox, 316Single replicate factorial, 568Small sample confidence interval, 119Specificity

sensitivity, 69, 76, 111, 689Split plot design, 609

ANOVA table, 612main plot error, 611, 626main plot SS, 612sub plot analysis, 76

Split-split plot design, 624–627an example, 640

798 Index

design layout, 624structure of ANOVA Table, 629

Square-root transformation, 430Analysis of Variance of, 469

Statistical tables, 777X2, 781Duncan’s multiple range test,

786, 787F distribution, 782, 783orthogonal polynomial

coefficients, 784standard normal, Z, 779t distribution, 780

Strip-plot designan example, 631design layout, 637structure of ANOVA table, 629

Standard Errorsfor the mean x, 166for the proportion p, 168for the difference of two means, 127for the difference of two

proportions, 127Student’s t distribution, 125, 138, 228Summation notation, 55Sums of squares

of contrasts, 415of orthogonal contrasts, 181from orthogonal polynomial

coefficients, 198in Yates’s algorithm, 542

Survival analysis, 719survival time, 720, 723, 732, 734,

740, 742Survival function, 720

computations, 732definition, 722probabilities, 725, 732

TTable of orthogonal polynomial

coefficients, 286t Table, 141, 452Tally, 44, 103Test of significance, 130, 131,

133, 519Test

of common intercept, 263of no interaction, 425of parallelism, 519for homogeneity of variances, 772for normality, 432lack of fit, 235

multiple comparisons, 175, 176Tests concerning two populations, 107

two sample Z test, 205two sample t test, 169, 174two proportions, 128

Test of independencein 2 × 2 contingency tables, 75

Test of Significance, 130, 131, 133,235, 634

Test statisticsCook’s, 251Pearson’s X2, 311, 312, 321, 353Likelihood Ratio test G2, 308Yates test, 542Mantel-Haenzel test, 317

Testing for a proportion, 141, 142The 23 factorial

ANOVA table, 539, 541calculating the SS, 438standard errors, 543

TransformationsArc Sine, 431–438Box-Cox, 438–440logarithmic, 431square root, 430, 431, 433, 439

Tree diagram, 69, 72Type I error, 133, 178Type II error, 133, 343Types of alternative hypotheses, 134

left-tailed, 134, 135right-tailed, 134two-sided, 136, 142, 243

t-test, 150, 155, 186Tukey’s test, 178–180

studentized range tables, 178

UUnbalanced one-way ANOVA, 171, 699Upper tail, 178

VVariate, 1, 3, 4, 42, 338, 503Variable

binary, 430categorical, 4, 28, 755concomitant, 503–505, 530continuous, 341, 680discrete, 103explanatory, 217, 237, 243, 251, 297,

697, 744independent, 243, 251, 255, 298, 531qualitative, 24, 194

Index 799

response, 194, 217, 218, 280, 295,355, 516, 662, 697, 757

Varianceanalysis, see Analysis of varianceof discrete random variable, 79heterogeneity, 343pooled, 148, 171, 186, 258, 759

Variationexplained or fitted, 226total, 8, 233, 346, 571

Venn diagram, 66Vertical factor, 632

WWeibull model, 736, 737Weighted Mean, 36–39

YYates’s algorithm, 409, 542–546Y-intercept, 217

Zz confidence intervals, 113z critical values, 155z score, 90z table, 779z test

for difference between means, 131,135, 207, 208, 210, 561

for difference betweenproportions, 127

for population mean, 423for population proportion, 128

Appendix: Statistical Tables

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