13.13
Kendallのタウ検定
Kendallのタウ検定はKendall順位係数検定とも呼ばれ、2 つの変数間の関連性を評価するためのノンパラメトリック手法です。この検定は、サンプルと母集団の分布が不明な場合に、有意な相関関係を識別するのに特に役立ちます。1938 年に英国の統計学者Maurice George Kendall卿に…
例えば、35個のヴィンテージ磁器のティーカップが、職人と一般の購入者によって独立してランク付けされている例を考えてみましょう。
ケンドールのタウ検定は、これら2つのランキングの間に関連性が存在するかどうかを調べるために使用できます。
ここで、帰無仮説は、職人のランキングと購入者のランキングとの間に相関関係がないことを示しています。対立仮説は、これら2つのランキングの間に相関関係が存在するというものです。
まず、たとえば職人のランキングごとに、データを特定の順序で並べ替えます。
これらのペアのランクは、カウントに変換する必要があります。
従来のカウント計算方法では、まずランクを特定し、同じ列のランクよりも多いランクの総数をカウントします。データ内のすべてのランクに対してこのプロセスを繰り返します。
検定統計量 tau は、次の式を使用して計算できます。
この量の重要性は、適切なコンピュータベースのツールを使用して取得できます。
ケンドールのタウは、データに同順位がない場合に、変数間の線形または非線形の単調な関連性を見つけるのに特に役立ちます。
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Q1: What is Kendall's tau test used for in statistics?
Kendall's tau test is a nonparametric method for assessing association between two variables when their distributions are unknown. It identifies whether a correlation exists between paired rankings without requiring strict distributional assumptions. The tau coefficient ranges from -1 to +1, where +1 indicates perfect positive correlation and -1 indicates perfect negative correlation, making it valuable for analyzing ordinal data.
Q2: How do you calculate the test statistic in Kendall's tau?
To calculate Kendall's tau, first arrange data by one variable's ranking. Then count how many ranks in the second column are higher than each rank position. Repeat this for all ranks to obtain counts. The test statistic tau is calculated using these counts in a specific equation. The significance of the resulting tau value can be determined using computer-based statistical tools.
Q3: What does a tau value of +1 or -1 indicate?
A tau value of +1 indicates that the ranks of two variables are perfectly similar, suggesting a strong positive correlation between them. Conversely, a tau value of -1 indicates that the ranks are perfectly dissimilar, suggesting a strong negative correlation. Positive tau values indicate positive relationships, while negative values signify negative relationships between the variables.
Q4: How does Kendall's tau compare to Spearman's rank correlation test?
Kendall's tau and Spearman's rank correlation test are equivalent and equally precise nonparametric methods for assessing rank correlation. Both tests are valid for analyzing associations between variables without strict distributional assumptions. However, Kendall's tau calculation is more straightforward when there are no ties in the data ranks and is more widely used for such data.
Q5: What are the null and alternative hypotheses in Kendall's tau test?
The null hypothesis states that no correlation exists between the two sets of rankings being compared. The alternative hypothesis is that a correlation does exist between the rankings. These hypotheses are tested by calculating the tau statistic and determining its significance, which indicates whether sufficient evidence exists to reject the null hypothesis.
Q6: When is Kendall's tau test most appropriate to use?
Kendall's tau test is particularly useful for finding linear or non-linear monotonic associations between variables when there are no ties in the data. It is valuable for analyzing ordinal data and exploring relationships without relying on strict assumptions about underlying distributions. The test is especially appropriate when sample and population distributions are unknown.
Q7: What is the rank intersection method in Kendall's tau calculation?
The rank intersection method involves arranging data into two columns with rankings from each variable, then drawing lines connecting matching ranks between columns. The total number of intersections formed by these lines is counted as X. This intersection count is then used in an equation to calculate N, which contributes to determining the tau coefficient and correlation strength.