Bivariate statistical measures

Bivariate statistical measures

1. Correlation

   • Definition: A statistical measure that evaluates the strength and direction of the relationship between two variables.
   • Types - methods: Pearson Spearman Kendall
     • Pearson: Measures the linear relationship between two continuous variables.
     • Spearman: Evaluates the monotonic (not necessarily linear) relationship between two variables.
     • Kendall: Calculates the agreement between the orders of two data sets.
   • Formula
   • Interpretation:
     • Value range: -1 to 1
     • r close to 1: positive relationship
     • r close to -1: negative relationship
     • r close to 0: weak relationship

2. Linear regression

   • Definition: A statistical model that describes the relationship between a dependent variable and one or more independent variables.
   • Types or methods: simple, multiple
     • Simple (an independent variable)
     • Multiple (more than one independent variable)
   • Coefficients:
     • Intercept (a): Value of the dependent variable when all independent variables are zero.
     • Coefficients (b): They represent the change in the dependent variable per unit change in the independent variable.
   • formula
   • Determination Coefficient(R^2)
     • Indicates the proportion of the variability of the dependent variable that can be explained by the regression.
     • Interpretation: R^2 close to 1 indicates good prediction ability.

3. Correlation Matrix

   • Definition: A table that shows the correlations between all possible combinations of variables in a data set.
   • Display:
     • Represented as a square matrix.
     • The element in row i and column j shows the correlation between the variables i and j.
   • Utility:
     • Identify patterns of association between variables.
     • It helps to understand how variables relate to each other.
     • It can be used for feature selection in multivariate analysis.
   • Interpretation of values ​​in the array:
     • The values ​​in the array vary between -1 and 1.
     • -1: Perfect negative correlation.
     • 1: Perfect positive correlation.
     • 0: Absence of correlation.
     • Values ​​close to -1 or 1 indicate a strong correlation.
     • Values ​​close to 0 indicate a weak or no correlation.

4. Residue Analysis

   • Definition: Evaluation of the differences between the observed values ​​and the values ​​predicted by the regression model.
   • Importance:
     • It allows verifying the validity of the regression model.
     • Identify patterns not captured by the model.
     • Helps detect possible violations of regression assumptions.
   • Methods:
     • Residue Charts:
       • Visualization of residuals (differences between observed and predicted values) as a function of the independent variables.
       • Non-random patterns in the graphs may indicate problems in the model.
     • Residue Histogram:
       • Graphic representation of the distribution of residuals.
       • It helps evaluate whether the residuals follow a normal distribution.
   • Interpretation:
     • Residuals close to 0 indicate that the model is predicting accurately.
     • Patterns in the graphs may suggest the presence of nonlinearity or heteroscedasticity (non-constant variance).
     • A histogram of residuals that resembles a normal distribution is desirable for statistical inference.

5. Atypical values

   • Definition: Outliers or unusual values ​​in a data set that differ significantly from the rest of the observations.
   • identification:
     • Visualizing data using boxplots, scatterplots, or statistical analysis such as the interquartile range rule.
     • Observation of values ​​that are outside specific ranges or that are statistically unusual
   • Effects on Regression and Correlation:
     • They can strongly influence the estimation of parameters in regressions.
     • They have the potential to distort the strength and direction of the correlation.
     • They can negatively affect the validity of statistical inferences.
   • Outliers Treatment:
     • Careful identification: Use graphical and statistical methods to confirm its presence.
     • Impact assessment: Determine if the elimination of outliers is justified and understand its impact on results.
     • Data transformation: Apply logarithmic or other transformations to reduce the influence of extreme values.
     • Imputation of values: Replace outliers with more representative values.
     • Robust models: Use robust methods that are less sensitive to the presence of outliers.

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