11.3
Berechnung und Interpretation des linearen Korrelationskoeffizienten
Der Korrelationskoeffizient r, der Anfang des 20. Jahrhunderts von Karl Pearson entwickelt wurde, ist numerisch und liefert ein Maß für die Stärke und…
Betrachten Sie den Datensatz des Kohlendioxidgehalts im Vergleich zur Jahrestemperatur über einen bestimmten Zeitraum. Das Streudiagramm der Datenpunkte zeigt ein wahrscheinliches lineares Muster zwischen den beiden Variablen.
Um ein geradliniges Muster zu bestätigen, wird der lineare Korrelationskoeffizient r berechnet.
Zuerst werden das Quadrat x, das Quadrat y und das Produkt aus x und y bestimmt und dann addiert. Die Anzahl der Datenpunkte beträgt 7.
Aus diesen Werten wird der Korrelationskoeffizient berechnet.
Die Bedeutung des Korrelationskoeffizientenwerts kann anhand der Tabelle mit dem kritischen Wert interpretiert werden.
Bei einem Signifikanzniveau von 0,05 und n gleich 7 ergibt sich ein kritischer Wert von 0,754.
Da der Modul von r größer als der kritische Wert ist, gibt es genügend Beweise für die Schlussfolgerung, dass es eine lineare Korrelation zwischen den Variablen gibt.
Der r-Quadrat-Wert zeigt an, dass 76,2 % der Variation der Jahrestemperatur durch die lineare Beziehung zwischen Kohlendioxidgehalt und Jahrestemperatur erklärt werden können.
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Q1: What does the linear correlation coefficient measure?
The linear correlation coefficient, r, is a numerical measure developed by Karl Pearson that quantifies the strength and direction of the linear association between two variables. It indicates whether variables move together in a predictable straight-line pattern. Values range from -1 to +1, where values closer to -1 or +1 indicate stronger linear relationships, while values near 0 suggest weak or no linear association.
Q2: How do you determine if a correlation coefficient is statistically significant?
Compare the absolute value of the calculated correlation coefficient to the critical value from a significance table at your chosen significance level. If the absolute value of r exceeds the critical value, the correlation is statistically significant. For example, at a 0.05 significance level with 7 data points, a critical value of 0.754 means any |r| greater than 0.754 indicates significant linear correlation between variables.
Q3: What is the coefficient of determination and what does it tell you?
The coefficient of determination, r², is the square of the correlation coefficient expressed as a percentage. It represents the percent of variation in the dependent variable that can be explained by the independent variable using the regression line. For instance, an r² of 76.2% means 76.2% of temperature variation is explained by carbon dioxide levels, while 23.8% remains unexplained.
Q4: What calculations are needed to compute the linear correlation coefficient?
To calculate r, you must first determine x², y², and the product of x and y for each data point, then sum these values. The correlation coefficient formula uses these sums along with the number of data points, n. These intermediate calculations provide the necessary components to quantify the linear relationship strength between your two variables.
Q5: Why is the linear correlation coefficient also called the Pearson product-moment correlation coefficient?
The coefficient is named after Karl Pearson, who developed it in the early 1900s. The term 'product-moment' refers to the calculation method, which involves computing products of paired values and their deviations from the mean. This naming convention distinguishes it from other correlation measures and honors its mathematical foundation.
Q6: What does it mean when the correlation coefficient falls between the critical values?
When r falls between the positive and negative critical values from the significance table, the correlation coefficient is not statistically significant. This means there is insufficient evidence to conclude a true linear relationship exists between the variables. In such cases, using the regression line for prediction is not recommended.
Q7: How does the unexplained variation relate to data scatter around the regression line?
The unexplained variation, calculated as 1 – r² and expressed as a percentage, represents the portion of dependent variable variation not explained by the regression line. This unexplained variation appears as the scattering of observed data points about the best-fit line, which can be examined using residual plots. Greater scatter indicates more unexplained variation and a weaker linear relationship.