Omnibus Tests in One Way Analysis of Variance Omnibus test

1 omnibus tests in 1 way analysis of variance

1.1 model assumptions in one-way anova
1.2 example

1.2.1 anova
1.2.2 dependent variable: time minutes respond
1.2.3 test of homogeneity of variances
1.2.4 dependent variable: time minutes respond


1.3 important remarks , considerations

omnibus tests in 1 way analysis of variance

the f-test in anova example of omnibus test, tests overall significance of model. significant f test means among tested means, @ least 2 of means different, result doesn t specify means different 1 other. actually, testing means differences made quadratic rational f statistic ( f=msb/msw). in order determine mean differ mean or contrast of means different, post hoc tests (multiple comparison tests) or planned tests should conducted after obtaining significant omnibus f test. may consider using simple bonferroni correction or other suitable correction. omnibus test can find in anova f test testing 1 of anova assumptions: equality of variance between groups. in one-way anova, example, hypotheses tested omnibus f test are:

h0: µ1=µ2=….= µk

h1: @ least 1 pair µj≠µj

these hypotheses examine model fit of common model: yij = µj + εij, yij dependant variable, µj j-th independent variable s expectancy, referred group expectancy or factor expectancy ; , εij errors results on using model.

the f statistics of omnibus test is:



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{\displaystyle f={\tfrac {{\displaystyle \sum _{j=1}^{k}n_{j}\left({\bar {y}}_{j}-{\bar {y}}\right)^{2}}/{(k-1)}}{{\displaystyle {\sum _{j=1}^{k}}{\sum _{i=1}^{n_{j}}}\left(y_{ij}-{\bar {y}}_{j}\right)^{2}}/{(n-k)}}}}

where,






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{\displaystyle {\bar {y}}}

overall sample mean,







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{\displaystyle {\bar {y}}_{j}}

group j sample mean, k number of groups , nj sample size of group j.

the f statistic distributed f(k-1,n-k),(α) under assuming of null hypothesis , normality assumption. f test considered robust in situations, when normality assumption isn t met.

model assumptions in one-way anova

random sampling.
normal or approximately normal distribution of in each group.
equal variances between groups.

if assumption of equality of variances not met, tamhane’s test preferred. when assumption satisfied can choose amongst several tests. although lsd (fisher’s least significance difference) strong test in detecting pairs of means differences, applied when f test significant, , less preferable since method fails in protecting low error rate. bonferroni test choice due correction suggested method. correction states if n independent tests applied α in each test should equal α /n. tukey’s method preferable many statisticians because control overall error rate. (more information on issue can found in anova book, such douglas c. montgomery’s design , analysis of experiments). on small sample sizes, when assumption of normality isn t met, nonparametric analysis of variance can made kruskal-wallis test, omnibus test example ( see following example ). alternative option use bootstrap methods assess whether group means different. bootstrap methods not have specific distributional assumptions , may appropriate tool use using re-sampling, 1 of simplest bootstrap methods. can extend idea case of multiple groups , estimate p-values.

example

a cellular survey on customers time-wait reviewed on 1,963 different customers during 7 days on each 1 of 20 in-sequential weeks. assuming none of customers called twice , none of them have customer relations among each other, 1 way anova run on spss find significant differences between days time-wait:

anova
dependent variable: time minutes respond

the omnibus f anova test results above indicate significant differences between days time-wait (p-value =0.000 < 0.05, α =0.05).

the other omnibus tested assumption of equality of variances, tested levene f test:

test of homogeneity of variances
dependent variable: time minutes respond

the results suggest equality of variances assumption can t made. in case tamhane’s test can made on post hoc comparisons.

some important remarks , considerations

a significant omnibus f test in anova procedure, in advance requirement before conducting post hoc comparison, otherwise comparisons not required. if omnibus test fails find significant differences between means, means no difference has been found between combinations of tested means. in such, protects family-wise type error, may increased if overlooking omnibus test. debates have occurred efficiency of omnibus f test in anova.

in paper review of educational research (66(3), 269-306) reviewed greg hancock, problems discussed:

william b. ware (1997) claims omnibus test significance required depending on post hoc test conducted or planned: ... tukey s hsd , scheffé s procedure one-step procedures , can done without omnibus f having significant. posteriori tests, in case, posteriori means without prior knowledge , in without specific hypotheses. on other hand, fisher s least significant difference test two-step procedure. should not done without omnibus f-statistic being significant.

william b. ware (1997) argued there number of problems associated requirement of omnibus test rejection prior conducting multiple comparisons. hancock agrees approach , sees omnibus requirement in anova in performing planned tests unnecessary test , potentially detrimental, hurdle unless related fisher s lsd, viable option k=3 groups.

other reason relating omnibus test significance when concerned protect family-wise type error.

this publication review of educational research discusses 4 problems in omnibus f test requirement:

first, in planned study, researcher s questions involve specific contrasts of group means while omnibus test, addresses each question tangentially , rather used facilitate control on rate of type error.

secondly, issue of control related second point: belief omnibus test offers protection not accurate. when complete null hypothesis true, weak family-wise type error control facilitated omnibus test; but, when complete null false , partial nulls exist, f-test not maintain strong control on family-wise error rate.

a third point, games (1971) demonstrated in study, f-test may not consistent results of pairwise comparison approach. consider, example, researcher instructed conduct tukey s test if alpha-level f-test rejects complete null. possible complete null rejected widest ranging means not differ significantly. example of has been referred non-consonance/dissonance (gabriel, 1969) or incompatibility (lehmann, 1957). on other hand, complete null may retained while null associated widest ranging means have been rejected had decision structure allowed tested. has been referred gabriel (1969) incoherence. 1 wonders if, in fact, practitioner in situation conduct mcp contrary omnibus test s recommendation.

the fourth argument against traditional implementation of initial omnibus f-test stems fact well-intentioned unnecessary protection contributes decrease in power. first test in pairwise mcp, such of disparate means in tukey s test, form of omnibus test itself, controlling family-wise error rate @ α-level in weak sense. requiring preliminary omnibus f-test amount forcing researcher negotiate 2 hurdles proclaim disparate means different, task range test accomplished @ acceptable α -level itself. if these 2 tests redundant, results of both identical omnibus test; probabilistically speaking, joint probability of rejecting both α when complete null hypothesis true. however, 2 tests not redundant; result joint probability of rejection less α. f-protection therefore imposes unnecessary conservatism (see bernhardson, 1975, simulation of conservatism). reason, , listed before, agree games (1971) statement regarding traditional implementation of preliminary omnibus f-test: there seems little point in applying overall f test prior running c contrasts procedures set [the family-wise error rate] α .... if c contrasts express experimental interest directly, justified whether overall f significant or not , (family-wise error rate) still controlled.