日: 2013年1月22日

How to execute multiple comparison

Student’s t-test would be executed to compare average between two groups. Then if you would like to compare average between 3 or more groups, what do you do? The test needs 2 steps process.

  1. Analysis of variance (ANOVA)
  2. Compare between each 2 groups

1. Analysis of variance

On analysis of variance, null hypothesis is that all groups belong to one population. Therefore, if null hypothesis has been rejected, all groups would not belong to one population.

If all groups belong to one population, the average of all groups, called as grand mean (MG), would be close to average of population. Furthermore, if all groups belong to one population, each average of each groups, for example, M1, M2, M3, would be close to grand mean. However, if any group doesn’t belong to one population, the average of other population would be far from MG. Then we need the indicator that represents how far each average of each groups from grand mean, corrected with number of sample, n. It is called as mean square among groups (MSA).

\displaystyle MSA = \frac{\sum_{i=1}^k n_i (M_i - MG)^2}{k-1}

MSA; mean square among groups. n; number of sample in each groups. i; number of group (incremental variable). k; number of groups.

Then calculate variances of each samples, correct with number of each sample and you would take index of variances in samples. It is mean square of error (MSE), average of variance in group.

\displaystyle MSE = \frac{\sum_{i=1}^k (n_i - 1)V_i}{\sum_{i=1}^{k}(n_i - 1)}

MSE; mean square of error. n; number of sample in each groups. i; number of group (incremental variable). k; number of groups. V; variance.

\displaystyle V = \frac{\sum(x - \bar x)^2}{n-1}

x; each value of samples in each groups. n; number of sample.

F statistics, calculated as ratio MSA to MSE, follows F distribution. When F statistics would be over a value, null hypothesis would be rejected and you could compare average between each groups.

\displaystyle F=\frac{MSA}{MSE}

2. Compare between each 2 groups

If null hypothesis would be rejected with ANOVA, you could compare between each groups with following method.

  • Bonferroni method
  • Tukey’s HSD
  • Dunnet’s procedure
  • Hsu’s MCB tests
  • Scheffe’s procedure

Bonferroni method may be easy to understand and use. Divided significance level \alpha by k, number of pairs, would be Bonferroni corrected significance level. See following chart.

Bonferroni Corrected Significance Level

多重比較するにはまず分散分析を行い，次いで各群間の比較を行う

　2 群間の平均値に差があるかを検定するには Student’s t 検定を行いました．今回は 3 群間の平均値に差があるかを検定する方法を述べます．検定は 2 段階に分けて行います．

  1. 分散分析 (ANOVA)
  2. 各群間の比較

1. 分散分析 (analysis of variance)

　分散分析では帰無仮説を『全ての群が同一の母集団に属する』とします．これを否定出来れば全ての群が同一母集団には属しないことが言えます．以下その方法を述べます．

　すべての群が同一母集団に属しているなら 3 群全部のサンプル平均値 (grand mean; MG) は母集団の平均値に近くなるはずです．さらにすべての群が同一母集団に属するなら，それぞれの群の平均値 (M1, M2, M3) は MG に近くなるはずです．逆に 3 群が異なる母集団に属するなら M1, M2, M3 は MG から離れた値になります．そこで各群の平均値が総平均値からどれだけ離れているか，それぞれの群のサンプル数 n で補正した指標を下記の式で表現します．これは平均値の群間差の平方和です．

\displaystyle MSA = \frac{\sum_{i=1}^k n_i (M_i - MG)^2}{k-1}

MSA; mean square among groups. n; number of sample in each groups. i; number of group (incremental variable). k; number of groups.

　次に各サンプルの分散を求め，各群のサンプル数で補正して 1 サンプルあたりのばらつきの指標とします．これは群内の分散の平均値となります．

\displaystyle MSE = \frac{\sum_{i=1}^k (n_i - 1)V_i}{\sum_{i=1}^{k}(n_i - 1)}

MSE; mean square of error. n; number of sample in each groups. i; number of group (incremental variable). k; number of groups. V; variance.

\displaystyle V = \frac{\sum(x - \bar x)^2}{n-1}

x; each value of samples in each groups. n; number of sample.

　下記の式のように MSA と MSE の比を取ると， MSA/MSE は F 分布に従います．F の値が一定以上となると帰無仮説は棄却され，全ての群が同一母集団には属しないことが言え，各群間の比較が可能となります．

\displaystyle F=\frac{MSA}{MSE}

2. 各群間の比較

　ANOVA の結果，全ての群が同一母集団には属しないことが証明された後に各群間を比較する方法にはいくつかあります．

  • Bonferroni 法
  • Tukey 法
  • Dunnet 法
  • Hsu’s MCB method
  • Scheffe’s procedure

　Bonferroni 法が分かりやすいので述べます．有意水準 \alpha を群数 k で除算した \alpha/k を有意水準とする方法です．下図は Bonferroni 法による有意水準を補正した場合としない場合とで有意水準がどう変化するか示したグラフです．

Bonferroni Corrected Significance Level