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3.4 Further Measures for Association Strength

Coefficient of Determination; r^2

  • The coefficient of determination gives a quantitative way of determining how much of the variation of the response variable is explained by variation in the explanatory variable.
  • It is represented by a lower-case r with a 2 superscript and can be calculated by squaring the correlation coefficient:

r^{2}=\left(\frac{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{(n-1) s_{x} s_{y}}\right)^{2}

  • When calculating the coefficient of determination, you will get a decimal answer, however, when interpreting the value, you should convert it into a percentage (multiply by 100).
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2.8 Non-Causal Relationships

Observed Association

  • The associations we find by collecting data and analysing are known as observed associations, as this is what we see.
  • It is worth noting an observed association does not necessarily mean there is an actual relationship between the two variables in question, or that their relationship is causal (as we will explore further in this topic).
  • An observed association may be the result of:
    • An actual relationship of some form between the variables.
    • Chance
    • Poor experimental design
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2.6 Pearson’s Correlation Coefficient

Meaning and Calculation

  • Pearson’s correlation coefficient provides a quantitative method for determining the strength and direction of a numerical association.
  • It is denoted by a lower-case r and can be calculated using the following formula:

r=\frac{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{(n-1) s_{x} s_{y}}

Where s_{x} and s_{y} are the standard deviations of the explanatory and response variables, respectively.

Limitations of using Pearson’s Correlation Coefficient

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2.5 Relationships between two Numerical Variables

Guidelines to Analysing Numerical Associations

  • Begin with context: what does the data represent?
  • Identify the explanatory and response variables.
  • Assess the form of the association: is it linear, non-linear or is there no association.
  • If it is linear, assess the strength (strong, moderate or weak). Ideally, do this using the Pearson’s correlation coefficient (detailed in 2.6 Pearson’s Correlation Coefficient), however if the raw data is not available, a qualitative assessment will suffice.
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2.1 Response and Explanatory Variables

Explanatory Variable

  • The explanatory variable (EV) is the variable used to explain or predict another variable (the response variable).
  • By convention, the explanatory variable is plotted along the x-axis of a graph, if it is numerical.

Response Variable

  • The response variable (RV) is the variable which is explained or predicted by the explanatory variable.
  • By convention, the response variable is plotted along the y-axis of a graph, if it is numerical.

Note: both explanatory and response variables can be either categorical or numerical variables.

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