????? ANalysis Of VAriance ANOVA

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????? ANalysis Of VArianceANOVA

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• pnhsieh_at_ntu.edu.tw

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??? ??? ??????
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?? ?? t-test12??? ANOVA2?(?)????
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• ANOVA???
• ??????????????(significant differences)
• ??????????,??ANOVA???????? t??(????????)??????t??(
  ????????????),????

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• T????????????????????????????????????,????????????
  ????????????,???????
• ????
• 1923?R.A. Fisher????????(analysis of variance,
  ANOVA)???????????????????????????????
• ???????????????????

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????????

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• ???????

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??????

• ???????????????????,?????????!
• 1920??40?????Sir Ronald A. Fisher,
  (1890-1962,??????)???
• The Design of Experiments, 1935
• ?????(analysis of vairance)?????lt????????IIgt??(194
  2)?????
• 3????????
• 10????
• 4???/??
• lt????????IVgt????(analysis of covariance)

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????ANOVA

• ?????????????,???????????(analysis of variance)
• ??,???????????????,??????(???)?????( variances )

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??????

• ?????????????,??????????????????????,?????????????
  ?????????????????????
• ????????????????(????)?????,????,??????

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(2-2)2 (3-2)2 (1-2)22
(2-4)2 (3-4)2 (1-4)2 (6-4)2 (7-4)2 (5-4)228
total SS (28) ??????? ??????(within-group
variability) (224) ?????????????
(28-(22)24).
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• SS Error and SS Effect
• the within-group variability (SS) is usually
  referred to as Error variance. This term denotes
  the fact that we cannot readily explain or
  account for it in the current design.
• the SS Effect we can explain. Namely, it is due
  to the differences in means between the groups.
  Put another way, group membership explains this
  variability because we know that it is due to the
  differences in means.

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ANOVA

• comparison of the variance
• between- groups variability (called Mean Square
  Effect, or MSeffect)
• within- group variability (called Mean Square
  Error, or Mserror)
• the null hypothesisthat there are no mean
  differences between groups in the population
• compare those two estimates of variance via the F
  test, which tests whether the ratio of the two
  variance estimates is significantly greater than
  1.

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T test vs. anova
  ExperimentalGroup 1 ExperimentalGroup 2
Males   231 675
Mean 2 6
Females   453 897
Mean 4 8

• total variance
• error (within-group) variability
• variability due to experimental group membership
• variability due to gender
• T test experiment or gender
• Anova experiment x gender
• interaction

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Main effects, two-way interaction
  Achievement-oriented Achievement-avoiders
Challenging TestEasy Test 105 510

• challenging tests make students work harder?
• achievement-oriented students work harder than
  achievement- avoiders?
• challenging tests make only achievement-oriented
  students work harder, while easy tests make only
  achievement- avoiders work harder

Main Effects
Interaction Effects
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Higher order interactions
Females Achievement-oriented Achievement-avoiders
Challenging TestEasy Test 105 510
Males Achievement-oriented Achievement-avoiders
Challenging TestEasy Test 16 61
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????????????

• ????(????)???(level)???,????????????
• ????????????
• ?????????????(level)
• ???????????,?2?level,1????,2????
• ?????????????,?4?level,???????????

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???????

• ??????(???)
• ??????????????(????)???????????,?????????????,????
  ??????????,????????????(one way ANOVA)
• ??????????????(????),????????,???????????(two way
  ANOVA)
• ??????????????(????),????????????,???????????(fact
  orial analysis of ANOVA)

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???????

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• ????
• ????

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???? ??? ???????
????? one way ANOVA ????? one way ANOVA ????? one way ANOVA
???? 1? One way ANOVA
???? 1? One way ANOVA (?????????)
????? two way ANOVA ????? two way ANOVA ????? two way ANOVA
?????? 2? two way ANOVA
?????? 2? two way ANOVA
??????? 1???,1??? two way ANOVA mixed design
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????????
???? ??? ???????
????? Factorial ANOVA ????? Factorial ANOVA ????? Factorial ANOVA
?????? 3?? 3 way ANOVA
?????? 3?? 3 way ANOVA
??????? ????,1??? 3 way ANOVA mixed design
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???? ANCOVA

• Analysis of Covariance
• ????????????,???????????,????????????????
• ????
• ?????(????)
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???? ??? ???????
??????? one way ANCOVA ??????? one way ANCOVA ??????? one way ANCOVA
???? 1???? 1??????? One way ANCOVA
???? 1? 1??????? One way ANCOVA (?????????)
??????? Factorial ANCOVA ??????? Factorial ANCOVA ??????? Factorial ANCOVA
?????? ????? 1??????? factorial ANCOVA
?????? ????? 1??????? factorial ANCOVA
??????? 1???? 1??????? factorial ANCOVA mixed design
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????????,MANOVA

• Multivariate analysis of variance
• ????????
• ?????(????),????????
• ?????(????),????????

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???? ??? ???????
???????? one way MANOVA ???????? one way MANOVA ???????? one way MANOVA
???? 1? One way MANOVA
???? 1? One way MANOVA (?????????)
???????? two way MANOVA ???????? two way MANOVA ???????? two way MANOVA
?????? 2? two way MANOVA
?????? 2? two way MANOVA
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???? ??? ???????
???????? Factorial MANOVA ???????? Factorial MANOVA ???????? Factorial MANOVA
?????? 3?? Factorial MANOVA
?????? 3?? Factorial MANOVA
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??????? MANCOVA

• Multivariate Analysis of Covariance

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????????one-Way ANOVA
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????????

• ?????????????????????
• ????????(one-way ANOVA),????????????????
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  ponse rate)
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• ??????????,????????,????????????????
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• ?2?????????????????(Q10)
• ????????????????(5 level ??)
• ??????(?? ??)

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????????

• ?????????????????????????????????
• ???? H0?1?2?3?k
• ???? ?1??2 ? ?3 ? ? ?k???????
• ??????,???????????,????,??????????????????????????
  ???????????????,???????????,?????????????

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???? vs. ????

• ??????????????????????????,??????
• ?????
• Variance between/among groups?
• ???????????????,??????????????
• Web??DM?????????
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• Variance within groups
• ???????????????,?????????,???????????,???????????
  ??,?????????????????????????
• ??????????(????????),?????????????????????????????
  ???????????????????????

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• Xij ??j?ij
• ?????Xij???3???
• Xij ??j????I???????
• ??????Xij????,?????????
• ?j?Xij??j????????????????,??j ?j-
  ?,????????????????,?????????(special
  effect)????????( treatment effect),???????????????
  ?????,?????????????????????????????,??????????????
  ??????????????????,???3??

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• ?ij??????????,??Xij???j???,??ij Xij - ?j
• ??, Xij??????????????
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???ANOVA???

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ANOVA???

• ????????ij?????
• ??(residual )??
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• ??????,???2?????
• SAS PROC Univariate?plot?normal????
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• ??????????????????????,??????????
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• ???????????,???????????,??????????????

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????????

• ??1
• ??? ???????????????(one-way ANOV)
• ????????
• ??2
• ???????????????
• ???????? ??????????????????

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??ANOVA???,????????p-value.435,????.05????(??)??
??,???????????????????????????
??ANOVA???,????????p-value.435,?????????.05????,?
???.435gt.05,????????,???????????????????????
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?? vs. ?????????

• ????????????????

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?????????????????????????????????????,????????????
??????????????????????????????????,??? Levene
?????????? ??,????????,??Post-hoc ???,??????????
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??????????,????????F??5.192, p??.000,?????????.05?
???,????.000lt.05,???????,?????????????????????????
??????????????????????(F5.192, df11, plt.001)
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????????,??????

• Post-hoc test
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Post-hoc test

• ??????,???????????????,post hoc
  ???????????????????????????
• ???????????????????
• ??????????????????,????????? ??????????????????,?
  alpha ??? .05?

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Post-hoc????????

• Tukey
• ???????????
• ?????????????
• Bonferroni t procedure
• ????????????,????
• ??Dunns procedure
• Scheffe???
• ????,????,????Scheffe?????????,??????????
• ???????????
• ????????????,????
• ????????(???????),?????????????,????????????,??Sch
  effe???????????

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Post-hoc????????

• Dunnett
• ?????
• ??????????????????????????,?web?email?DM??????????
  ,???????????????????

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• ???????,???????????????,?????????????????,????????
  ???????,??????????????????????????

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????
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ANOVA

• ?????????????????
• ????????????????????????????????????
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• ???????????,????????
• ??????
• ?????????????????
• ??????????????(??????) ????(?????)

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ANOVA????

• ?????????????,????????,????????????????ANOVA??????
  ?
• One-way ANOVA??????????,??????
• Play It Safe Table
• Degree of freedom
• Sums of squares
• Mean squares
• F ratios

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Example 1

• Independent variable
• Coworker program????3??,????
• Consultant program??????3??,????
• Self-program?????3?
• Dependent variable
• Job performance7???,???????
• ???40???

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????????,?????
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Example 2

• Independent variable
• Coworker program????3??,????
• Consultant program????3??,????
• Self-program?????3?
• ???3000???
• Dependent variable7
• Job performance7???,???????
• Organizational commitment
• Job commitment
• Job satisfaction
• Turnover intention
• Job stress
• Role ambiguity

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??????????
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??7?ANOVA?? ?Play It Safe table
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????????????????,?F?????
??7?ANOVA?? ?Play It Safe table
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????????ANOVA F ratios
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