Abstract
When the two different methods of treatment are to be compared for effectiveness, preliminary information is in many cases obtained by experiments on laboratory animals. Suppose we are interested in two ointment preparations. The question arises: Does there or does there not exist a difference in the effectiveness of the two preparations? There are test animals at our disposal on which we can produce the seat of a disease. Let the measure of effectiveness be the amount oftime required for recovery:
-
1.
The simplest approach would be to divide a group of test animals randomly into two subgroups of equal size, treat one group by method one and the other by method two, and then compare the results of the therapies.
-
2.
The following approach is more effective: Test animals are paired in such a way that the individual pairs are as homogeneous as possible with regard to sex, age, weight, activity, etc. The partners are then assigned randomly (e.g., by tossing a coin) to the two treatments. The fact that the experimenter hardly ever has a completely homogeneous collection of animals at his disposal is taken into account in this procedure.
-
3.
The following procedure is considerably more effective: A group of test animals is chosen and a so-called right-left comparison carried out. That is, we produce on the right and left flank of each individual (or any such natural homogeneous subgroup of size two, like a pair of twins or the two hands of the same person) two mutually independent seats of a disease, and allot the two treatments to the two flanks, determining by a random process which is to be treated by the one method and which by the other (cf., also Section 7.7).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
[8:4] Chapter 4
Adler, F.: Yates correction and the statisticians. J. Amer. Statist. Assoc. 46 (1951), 490–501 [see also 47 (1952), 303 and American Statistician 30 (1976), 103–104]
Bateman, G.: On the power function of the longest run as a test for randomness in a sequence of alternatives. Biometrika 35 (1948), 97–112 [see also 34 (1947), 335–339; 44 (1957), 168–178; 45 (1958), 253–256; 48 (1961), 461–465]
Bennett, B. M.: Tests of hypotheses concerning matched samples. J. Roy. Statist. Soc. B 29 (1967), 468–474.
Bennett, B. M.: On tests for order and treatment differences in a matched 2×2. Biometrische Zeitschr. 13 (1971), 95–99
Bennett, B. M., and Horst, C.: Supplement to Tables for Testing Significance in a 2 × 2 Contingency Table. New York 1966
Bennett, B. M., and Hsu, P.: On the power function of the exact test for the 2 × 2 contingency table. Biometrika 47 (1960), 393–397 [editorial note 397, 398, correction 48 (1961), 475]
Bennett, B. M., and Underwood, R. E.: On McNemar’s test for the 2 × 2 table and its power function. Biometrics 26 (1970), 339–343 [see also 27 (1971), 945–959 and Psychometrika 47 (1982), 115–118]
Berchtold, W.: Die Irrtumswahrscheinlichkeiten des χ 2-Kriteriums für kleine Versuchszahlen. Z. angew. Math. Mech. 49 (1969), 634–636
Berkson, J.: In dispraise of the exact test. Do the marginal totals of the 2 × 2 table contain relevant information respecting the table proportions? Journal of Statistical Planning and Inference 2 (1978), 27–42
Bihn, W. R.: Wandlungen in der statistischen Zeitreihenanalyse und deren Bedeutung für die ökonomische Forschung. Jahrb. Nationalök. Statistik 180 (1967), 132–146 (see also Parzen 1967 and Nullau 1968)
Birnbaum, Z. W.: Numerical tabulation of the distribution of Kolmogorov’s statistic for finite sample size. J. Amer. Statist. Assoc. 47 (1952), 425–441
Blyth, C. R., and Hutchinson, D. W.: Table of Neyman-shortest unbiased confidence intervals for the binomial parameter. Biometrika 47 (1960), 381–391
Blyth, C. R., and Still, H. A.: Binomial confidence intervals. J. Amer. Statist. Assoc. 78 (1983), 108–116
Bogartz, R. S.: A least squares method for fitting intercepting line segments to a set of data points. Psychol. Bull. 70 (1968), 749–755 (see also 75 [1971], 294–296)
Box, G. E. P., and Jenkins, G. M.: Time Series Analysis, Forecasting and Control. (Holden-Day, pp. 575) San Francisco 1976
Bradley, J. V.: A survey of sign tests based on the binomial distribution. J. Qual. Technol. 1 (1969), 89–101
Bredenkamp, J.: F-Tests zur Prüfung von Trends und Trendunterschieden. Z. exper. angew. Psychologie 15 (1968), 239–272
Bross, I. D. J.: Taking a covariable into account. J. Amer. Statist. Assoc. 59 (1964), 725–736
Casagrande, J. T., Pike, M. C., and Smith, P. G.: The power function of the “exact” test for comparing two binomial distributions. Applied Statistics 27 (1978), 176–180
Clopper, C. J., and Pearson, E. S.: The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika 26 (1934), 404–413
Cochran, W. G.: The comparison of percentages in matched samples. Biometrika 37 (1950), 256–266.
Cochran, W. G.: The χ 2-test of goodness of fit. Ann. Math. Statist. 23 (1952), 315–345 [see Applied Statistics 29 (1980), 292–298 and Biometrika 67 (1980), 447–453].
Cochran, W. G.: Some methods for strengthening the common chi-square tests. Biometrics 10 (1954), 417–451
Cochran, W. G.: Sampling Techniques, 2nd edition, J. Wiley, New York, 1963
Conover, W. J.: A Kolmogorov goodness-of-fit test for discontinuous distributions. J. Amer. Statist. Assoc. 67 (1972), 591–596
Cox, D. R., and Stuart, A.: Some quick sign tests for trend in location and dispersion. Biometrika 42 (1955), 80–95 [cf., 55 (1968), 381–386; 67 (1980), 375–379]
Crow, E. L.: Confidence intervals for a proportion. Biometrika 43 (1956), 423–435
Crow, E. L., and Gardner, R. S.: Confidence intervals for the expectation of a Poisson variable. Biometrika 46 (1959), 441–453
Croxton, F. E., and Cowden, D. J.: Applied General Statistics. 2nd ed. (Prentice-Hall) New York 1955
Csorgo, M., and Guttman, I.: On the empty cell test. Technometrics 4 (1962), 235–247
Cureton, E. E.: The normal approximation to the signed-rank sampling distribution when zero differences are present. J. Amer. Statist. Assoc. 62 (1967), 1068–1069 [see also 69 (1974), 368–373]
Darling, D. A.: The Kolmogorov-Smirnov, Cramér-von Mises tests. Ann. Math. Statist. 28 (1957), 823–838
David, F.N.: A χ 2 ‘smooth’ test for goodness of fit. Biometrika 34 (1947), 299–310.
David, F.N.: Two combinatorial tests of whether a sample has come from a given population. Biometrika 37 (1950), 97–110
David, H. A., Hartley, H. O., and Pearson, E. S.: The distribution of the ratio, in a single normal sample, of range to standard deviation. Biometrika 41 (1954), 482–493
Davis, H. T.: The Analysis of Economic Time Series. San Antonio, Texas 1963
Dixon, W. J., and Mood, A. M.: The statistical sign test. J. Amer. Statist. Assoc. 41 (1946), 557–566 [see Int. Statist. Rev. 48 (1980), 19–28]
Documenta Geigy: Wissenschaftliche Tabellen, 7th edition, Basel 1968, pages 85–103 and 109–123 (8th revised edition 1980)
Duckworth, W. E., and Wyatt, J. K.: Rapid statistical techniques for operations research workers. Oper. Res. Quarterly 9 (1958), 218–233
Dunn, J. E.: A compounded multiple runs distribution. J. Amer. Statist. Assoc. 64 (1969), 1415–1423
Eisenhart, C., Hastay, M. W., and Wallis, W. A.: Techniques of Statistical Analysis. New York 1947
Feldman, S. E., and Klinger, E.: Short cut calculation of the Fisher-Yates “exact test”. Psychometrika 28 (1963), 289–291
Finkelstein, J. M., and Schafer, R. E.: Improved goodness-of-fit tests. Biometrika 58 (1971), 641–645
Finney, D. J., Latscha, R., Bennett, B. M., and Hsu, P.: Tables for Testing Significance in a 2 × 2 Contingency Table. Cambridge 1963
Gail, M., and Gart, J. J.: The determination of sample sizes for use with the exact conditional test in 2 × 2 comparative trials. Biometrics 29 (1973), 441–448 [see Haseman (1978)]
Gart, J. J.: Approximate confidence limits for the relative risk. Journal of the Royal Statistical Society B 24 (1962), 454–463, 458 [for odds ratio and relative risk see Amer. J. Epidemiology 115 (1982), 453–470 and 118 (1983), 396–407].
Gart, J. J.: An exact test for comparing matched proportions in crossover designs. Biometrika 56 (1969), 75–80 [see also Biometrics 27 (1971), 945–959 and Rev. Int. Stat. Inst. 39 (1971), 148–169].
Gart, J. J.: The analysis of ratios and cross product ratios of Poisson variates with application to incidence rates. Commun. Statist.-Theory and Methods A 7 (1978), 917–937
Gebhardt, F.: Verteilung und Signifikanzschranken des 3. und 4. Stichprobenmomentes bei normal-verteilten Variablen. Biometrische Zeitschr. 8 (1966), 219–241
Gildemeister, M., and Van der Waerden, B. L.: Die Zulässigkeit des /2-Kriteriums für kleine Versuchszahlen. Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Nat. Kl. 95 (1944), 145–150
Glasser, G. J.: A distribution-free test of independence with a sample of paired observations. J. Amer. Statist. Assoc. 57 (1962), 116–133
Good, I. J.: Significance tests in parallel and in series J. Amer. Statist. Assoc. 53 (1958), 799–813 [see also Biometrics 31 (1975), 987–992]
Grizzle, J. E.: Continuity correction in the χ 2-test for 2 × 2 tables. The American Statistician 21 (Oct. 1967), 28–32 [as well as 23 (April 1969), 35; cf. J. Amer. Statist. Assoc. 69 (1974), 374–382]
Harris, B. (Ed.): Spectral Analysis of Time Series. (Wiley, pp. 319) New York 1967
Hart, B. I.: Significance levels for the ratio of the mean square successive difference to the variance. Ann. Math. Statist. 13 (1942), 445–447
Haseman, J. K.: Exact sample sizes for use with the Fisher-Irwin test for 2 × 2 tables. Biometrics 34 (1978), 106–109 [see Fleiss, J. L., Tytun, A., and Ury, H. K.: Biometrics 36 (1980), 343–346 and 347–351]
See Fleiss, J. L., Tytun, A., and Ury, H. K.: Biometrics 36 (1980), 343–346 and 347–351
Jenkins, G. M.: Spectral Analysis and Its Applications. (Holden-Day, pp. 520) San Francisco 1968
Jenkins, G. M., and Watts, D. E.: Spectrum Analysis and Its Applications. (Holden-Day, pp. 350) San Francisco 1968
Jesdinsky, H. J.: Orthogonale Kontraste zur Prüfung von Trends. Biometrische Zeitschrift 11 (1969), 252–264
Johnson, E. M.: The Fisher-Yates exact test and unequal sample sizes. Psychometrika 37 (1972), 103–106 [see also Applied Statistics 28 (1979), 302]
Kincaid, W. M: The combination of tests based on discrete distributions. J. Amer. Statist. Assoc. 57 (1962), 10–19 [see also 66 (1971), 802–806 and 68 (1973), 193–194]
Klemm, P. G.: Neue Diagramme für die Berechnung von Vierfelderkorrelationen. Biometrische Zeitschr. 6 (1964), 103–109
Kolmogorov, A.: Confidence limits for an unknown distribution function. Ann. Math. Statist. 12 (1941), 461–463
Koziol, J. A., and Perlman, M. D.: Combining independent chi-squared tests. Journal of the American Statistical Association 73 (1978), 753–763
Kruskal, W. H.: A nonparametric test for the several sample problem. Ann. Math. Statist. 23 (1952), 525–540
Kullback, S., Kupperman, M., and Ku, H. H.: An application of information theory to the analysis of contingency tables, with a table of 2n ln n, n = 1 (1) 10,000. J. Res. Nat. Bur. Stds. B 66 (1962), 217–243
Le Roy, H. L.: Ein einfacher χ 2-Test für den Simultanvergleich der inneren Struktur von zwei analogen 2×2- Häufigkeitstabellen mit freien Kolonnen- und Zeilen-totalen. Schweizer. landw. Forschg. 1 (1962), 451–454
Levene, H.: On the power function of tests of randomness based on runs up and down. Ann. Math. Statist. 23 (1952), 34–56
Li, J. C. R.: Statistical Inference. Vol. I (Edwards Brothers, pp. 658) Ann Arbor, Mich. 1964, p. 466
Lienert, G. A.: Die zufallskritische Beurteilung psychologischer Variablen mittels verteilungsfreier Schnelltests. Psychol. Beiträge 7 (1962), 183–215
Lillefors, H. W.: On the Kolmogorov-Smirnov test for normality with mean and variance unknown. J. Amer. Statist. Assoc. 62 (1967), 399–402, Corrigenda 64 (1969), 1702.
Lillefors, H. W.: On the Kolmogorov-Smirnov test for the exponential distribution with mean unknown. J. Amer. Statist. Assoc. 64 (1969), 387–389 [see also Biometrika 63 (1976), 149–160]
Ludwig, O.: Über die stochastische Theorie der Merkmalsiterationen. Mitteilungsbl. math. Statistik 8 (1956), 49–82
MacKinnon, W. J.: Table for both the sign test and distribution-free confidence intervals of the median for sample sizes to 1, 000. J. Amer. Statist. Assoc. 59 (1964), 935–956
Makridakis, S.: A survey of time series. International Statistical Review 44 (1976), 29–70.
Makridakis, S.: Time-series analysis and forecasting: an update and evaluation. International Statistical Review 46 (1978), 255–278 [see J. Roy. Statist. Soc. A 142 (1979), 97–145]
Marascuilo, L. A., and McSweeney, Maryellen: Nonparametric post hoc comparisons for trend. Psychological Bulletin 67 (1967), 401–412 [see also 92 (1982), 517–525]
Massey, F. J. Jr.: The Kolmogorov-Smirnov test for goodness of fit. J. Amer. Statist. Assoc. 46 (1951). 68–78 [see also Allgem. Stat. Arch. 59 (1975), 228–250]
Maxwell, A. E.: Analysing Qualitative Data. 2nd edition. (Methuen) London 1970
McCornack, R. L.: Extended tables of the Wilcoxon matched pair rank statistic. J. Amer. Statist. Assoc. 60 (1965), 864–871 [see also 65 (1970), 974–975, 69 (1974), 255–258, 368–373 and Method. Inform. Med. 14 (1975), 224–230]
McNemar, Q.: Note on sampling error of the differences between correlated proportions or percentages. Psychometrika 12 (1947), 153–154
Miller, L. H.: Table of percentage points of Kolmogorov statistics. J. Amer. Statist. Assoc. 51 (1956), 111–121
Moore, P. G.: The properties of the mean square successive difference in samples from various populations. J. Amer. Statist. Assoc. 50 (1955), 434–456
Neumann, J. von, Kent, R. H., Beilinson, H. B., and Hart, B. I.: The mean square successive difference. Ann. Math. Statist. 12 (1941), 153–162
Nicholson, W. I.: Occupancy probability distribution critical points. Biometrika 48 (1961), 175–180
Nullau, B.: Verfahren zur Zeitreihenanalyse. Vierteljahreshefte zur Wirtschaftsforschung, Berlin 1968, 1, 58–82 (see DIW-Beitr. z. Strukturf., H. 7/1969; Wirtsch. u. Stat. H. 1/1973, H. 2 and 5/1975)
Olmstead P. S.: Runs determined in a sample by an arbitrary cut. Bell Syst. Techn. J. 37 (1958), 55–82
Ott, R. L., and Free, S. M.: A short-cut rule for a one-sided test of hypothesis for qualitative data. Technometrics 11 (1969), 197–200
Parzen, E.: The role of spectral analysis in time series analysis. Rev. Int. Statist. Inst. 35 (1967), 125–141 (cf. Empirical Time Series Analysis. Holden-Day, San Francisco, Calif. 1969)
Patnaik, P. B.: The power function of the test for the difference between two proportions in a 2 × 2 table. Biometrika 35 (1948), 157–175
Paulson, E., and Wallis, W. A.: Planning and analyzing experiments for comparing two percentages. In Eisenhart, Ch., M. W. Hastay and W. A. Wallis (Eds.), Selected Techniques of Statistical Analysis, McGraw-Hill, New York and London 1947, Chapter 7
Pearson, E. S.: Table of percentage points of √ b 1 and b 2 in normal samples; a rounding off. Biometrika 52 (1965), 282–285
Pearson, E. S., and Hartley, H. O.: Biometrika Tables for Statisticians. Vol. I, 3rd ed., Cambridge 1966, 1970
Pearson, E. S., and Stephens, M. A.: The ratio of range to standard deviation in the same normal sample. Biometrika 51 (1964), 484–487
Plackett, R. L.: The continuity correction in 2 × 2 tables. Biometrika 51 (1964), 327–337 [see Biometrical Journal 22 (1980), 241–248]
Quandt, R. E.: Statistical discrimination among alternative hypotheses and some economic regularities. J. Regional Sci. 5 (1964), 1–23.
Quandt, R. E.: Old and new methods of estimation and the Pareto distribution. Metrika 10 (1966), 55–82
Radhakrishna, S.: Combination of results from several 2 × 2 contingency tables. Biometrics 21 (1965), 86–98
Rao, C. R.: Linear Statistical Inference and Its Applications. 2nd ed. (Wiley) New York 1973, pp. 404–10
Rehse, E.: Zur Analyse biologischer Zeitreihen. Elektromedizin 15 (1970), 167–180
Rhoades, H. M., and Overall, J. E.: A sample size correction for Pearson chi-square in 2 × 2 contingency tables. Psychological Bulletin 91 (1982), 418–428
Runyon, R. P., and Haber, A.: Fundamentals of Behavioral Statistics. (Addison-Wesley, pp. 304) Reading, Mass. 1967, p. 258
Sachs, L.: Statistische Methoden. 6th revised edition (Springer, 133 pages) Berlin, Heidelberg, New York 1984, pages 69–70, 72–75.
Sachs, L.: Numerischer Vergleich von 11 Konkurrenten des klassischen Vierfelder-χ 2-Tests bei kleinem Stichprobenumfang. Habilitationsschrift, Kiel 1974
Sandler, J.: A test of the significance of the difference between the means of correlated measures, based on a simplification of Student’s t. Brit. J. Psychol. 46 (1955), 225–226
Sarris, V.: Nichtparametrische Trendanalysen in der klinisch-psychologischen Forschung. Z. exper. angew. Psychologie 15 (1968), 291–316
Seeger, P.: Variance analysis of complete designs: Some practical aspects. (Almqvist and Wiksell, pp. 225) Uppsala 1966, pp. 166–190
Seeger, P., and Gabrielsson, A.: Applicability of the Cochran Q test and the F test for statistical analysis of dichotomous data for dependent samples. Psychol. Bull. 69 (1968), 269–277
Shapiro, S. S., and Wilk, M. B.: An analysis of variance test for normality (complete samples). Biometrika 52 (1965), 591–611.
Shapiro, S. S., and Wilk, M. B.: Approximations for the null distribution of the W statistic Technometrics 10 (1968), 861–866 [cf. Statist. Neerl. 22 (1968), 241–248 and 27 (1973), 163–169]
Shapiro, S. S., Wilk, M. B., and Chen, H. J.: A comparative study of various tests for normality. J. Amer. Statist. Assoc. 63 (1968), 1343–1372 [cf. 66 (1971), 760–762 and 67 (1972), 215–216]
Slakter, M. J.: A comparison of the Pearson chi-square and Kolmogorov goodness-of-fit tests with respect to validity. J. Amer. Statist. Assoc. 60 (1965), 854–858; Corrigenda : 61 (1966), 1249 [cf. 69 (1974), 730–737 and 71 (1976), 204–209]
Smirnov, N.: Tables for estimating the goodness of fit of empirical distributions. Ann. Math. Statist. 19 (1948), 279–281 [cf. J. Roy. Statist. Soc. 38 (1976), 152–156]
Stephens, M. A.: Use of the Kolmogorov-Smirnov, Cramér-Von Mises and related statistics without extensive tables. J. Roy. Statist. Soc. B32 (1970), 115–122
Stevens, W. L.: Distribution of groups in a sequence of alternatives. Ann. Eugenics 9 (1939), 10–17.
Stevens, W. L.: Accuracy of mutation rates. J. Genetics 43 (1942), 301–307
“Student” (W. S. Gosset): The probable error of a mean. Biometrika 6 (1908), 1–25
Suits, D. B.: Statistics: An Introduction to Quantitative Economic Research. Chicago, Ill. 1963, Chapter 4
Swed, Frieda, S., and Eisenhart, C.: Tables for testing randomness of grouping in a sequence of alternatives. Ann. Math. Statist. 14 (1943), 83–86
Tate, M. W., and Brown, Sara, M.: Note on the Cochran Q-test. J. Amer. Statist. Assoc. 65 (1970), 155–160 [see also 68 (1973), 989–993; 70 (1975), 186–189; 72 (1977), 658–661; Biometrics 21 (1965), 1008–1010 and 36 (1980), 665–670]
Thomson, G. W.: Bounds for the ratio of range to standard deviation. Biometrika 42 (1955), 268–269
Tukey, J. W., and McLaughlin, D. H.: Less vulnerable confidence and significance procedures for location based on a single sample: Trimming/Winsorization. Sankhya Ser. A 25 (1963), 331–352
Ury, H. K.: A note on taking a covariable into account. J. Amer. Statist. Assoc. 61 (1966), 490–495
Vessereau, A.: Sur les conditions d’application du criterium χ 2 de Pearson. Bull. Inst. Int. Statistique 36 (3) (1958), 87–101
Waerden, B. L. van der: Mathematical Statistics. Springer-Verlag, New York 1969
Wallis, W. A.: Rough-and-ready statistical tests. Industrial Quality Control 8 (1952) (5), 35–40
Wallis, W. A., and Moore, G. H.: A significance test for time series analysis. J. Amer. Statist. Assoc. 36 (1941), 401–409
Walter, E.: Über einige nichtparametrische Testverfahren. I, II. Mitteilungsbl. Mathemat. Statistik 3 (1951), 31–44, 73–92.
Walter, E.: χ 2-Test zur Prüfung der Symmetric bezùglich Null. Mitteilungsbl. Mathemat. Statistik 6 (1954), 92–104.
Walter, E.: Einige einfache nichtparametrische überall wirksame Tests zur Prüfung der Zweistich-probenhypothese mit paarigen Beobachtungen. Metrika 1 (1958), 81–88
Weichselberger, K.: Über eine Theorie der gleitenden Durchschnitte und verschiedene Anwendungen dieser Theorie. Metrika 8 (1964), 185–230
Wilcoxon, F., Katti, S. K., and Wilcox, Roberta A.: Critical Values and Probability Levels for the Wilcoxon Rank Sum Test and the Wilcoxon Signed Rank Test. Lederle Laboratories, Division Amer. Cyanamid Company, Pearl River, New York, August 1963
Wilcoxon, F., Katti, S. K., and Wilcox, Roberta A.: Some Rapid Approximate Statistical Procedures. Lederle Laboratories, Pearl River, New York 1964
Wilk, M. B., and Shapiro, S. S.: The joint assessment of normality of several independent samples. Technometrics 10 (1968), 825–839 [see also J. Amer. Statist. Assoc. 67 (1972), 215–216]
Woolf, B.: The log likelihood ratio test (the G-test). Methods and tables for tests of heterogeneity in contingency tables. Ann. Human Genetics 21 (1957), 397–409
Yamane, T.: Statistics: An Introductory Analysis. 2nd ed. (Harper and Row, pp. 919) New York 1967, pp. 330–367, 845–873
Yates, F.: Contingency tables involving small numbers and the χ 2-text. Supplement to the Journal of the Royal Statistical Society 1 (1934), 217–235 [cf. Biometrika 42 (1955), 404–411 and Biometrical Journal 22 (1980), 241–248]
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer-Verlag New York Inc.
About this chapter
Cite this chapter
Sachs, L. (1984). Further Test Procedures. In: Applied Statistics. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5246-7_7
Download citation
DOI: https://doi.org/10.1007/978-1-4612-5246-7_7
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-9755-0
Online ISBN: 978-1-4612-5246-7
eBook Packages: Springer Book Archive