4:Chapter 4: Regression Analysis

Biostatistics & Statistical Analysis - Chapter 4: Relationships: Regression

• This chapter focuses on regression analysis, specifically the least-squares regression line.

• Previous learning objectives involve understanding bivariate data, scatterplots, interpreting scatterplots, and incorporating categorical variables.

• Correlation coefficient 'r' plays a role in demonstrating relationships.

• Key learning objectives for regression include: the least-squares regression line, facts about least-squares regression, outliers and impactful observations, working with logarithmic transformations, cautions about correlation and regression, and understanding association does not equal causation

• The least-squares regression line minimizes the sum of squared vertical distances from data points to the line.

• Residuals are the vertical distances from each data point to the least-squares regression line. The sum of all residuals equals zero.

• Notation for regression analysis introduces variables like 'ŷ' as the predicted value on the regression line, which is a linear equation 'ŷ = a + bx.'

• Parameters within the equation 'ŷ = a + bx' include 'a' (intercept) and 'b' (slope). Specific calculator/software notations vary.

• Interpreting the regression line introduces slope as the expected average change in 'y' for each unit change in 'x' and the intercept as a mathematical descriptor, not necessarily a property of the data itself.

• Finding the line involves calculation: slope (b) = r (correlation coefficient) * (standard deviation of y) / (standard deviation of x). Intercept (a) = mean of y - slope (b) * mean of x.

• Plotting the line requires two different 'x' values, using the regression equation to calculate the respective 'y' values and plotting the line through these points.

• Regression line always passes through mean x and mean y values.

• Important facts about least-squares regression: distinction between explanatory and response variables, slope proportionality to correlation between variables, and regression line passing through (mean x, mean y)

• Correlation coefficient measures the strength of association, while the square of the correlation (r²) represents the fraction of variance in y attributable to the regression model.

• Interpreting r² values: low r² values denote weak association, high values signify strong association (approaching 1). Nearly half the variation in y is clarified as r²=0.49 (or 49%).

• Outliers and influential observations are points deviating from overall patterns. Influential points markedly change the regression if removed.

• Logarithmic transformations are used for skewed data (either right or left skewed). Logarithmic transformations are made to the response variable, followed by regression analysis, and results must be re-transformed back to original units.

• Making predictions: Using regression equation for 'y' within the observed data range (interpolation). Extracting data outside the observed data range is extrapolation and should be avoided.

• Association does not equal causation: External causes or confounding variables (lurking variables) can create associations. Two variables are confounded when their effects on the response variable cannot be distinguished.

• Examples of lurking variables (external factors) in relationship analyses are illustrated with examples involving shoe size/reading skills in children, wine consumption/heart disease, and the relation between per capita chocolate consumption and Nobel laureates to illustrate how a lurking variable could be a confound.

• Establishing causation involves: strong associations, consistency, higher doses correlating with more significant responses, cause preceding effect, and plausible mechanisms.

• Boxplots visually summarize data location and spread on the interval scale (either discrete or continuous data). These are also useful to determine patterns of skewing of the data or identify if any outliers are present.

• Histograms are useful as summary statistics to graph out continuous data by illustrating its location, central value, and variability. Histograms can be used to identify outliers or gaps in the data set. Histograms can be misleading if they are not constructed properly.

• Quantile-quantile plots indicate whether datasets are normally distributed or not and can determine the distribution type.

Explore Chapter 4 of Biostatistics focusing on regression analysis, including the least-squares regression line and correlation coefficient 'r'. This chapter covers key objectives like outliers, logarithmic transformations, and understanding the nuances of correlation and causation. Test your knowledge of these crucial concepts in statistical relationships.