Forensic Statistics AFTER MIDTERM

Slope of linear regression

tells us how much additional stretching we predict per extra “session” (“independent trial”)

Simple Linear Regression

studies the relationship between a response variable (y) and a single explanatory variable (x)

• describes the existence of a linear relationship between the two variables

• a continuous response variable y and one quantitative explanatory variable x (might be continuous or discrete)

  • x = explanatory, independent, cause

  • y = response, dependent, effect

How do we choose the slope and intercept for simple linear regression?

we estimate the population parameters from our measurements

• for any candidate: Residual = Observed Value - Predicted Value

• we choose the intercept and slope that make the sum of squared residuals as small as possible = Least Squares

How do we calculate the slope and intercept of the best-fitting line for simple linear regression?

• TOP = when play sessions are above average, is sock stretching also above average? The products measure the pattern

• BOTTOM = measures how spread out the number of play sessions is

• RATIO = gives slope

Equation for intercept in linear regression best-fitting line

Independence

errors from different observations are independent

• checked mainly by looking at how we collected the data

• Are observations linked or repeatedly measured?

Linearity

the mean response follows a straight-line relationship with x

• checking residuals → curved = bad

Constant Variance

the spread around the regression line is similar across x

Heteroscedasticity

when the residual data resembles a funnel-shape; not a good fit!

• when finding constant variance

Normality

errors are approximately normally distributed around zero — for the usual confidence intervals and tests

What does a scatterplot inform you about?

1. The form

2. The direction

3. The strength

all of the relationship between x and y

• we can fit a line, but that doesn’t mean the relationship is strong

Pearson Correlation Coefficient, r

measure of the direction and strength of the linear relationship between 2 quantitative variables

• + = larger x tends to accompany larger y

• - = larger x tends to accompany smaller y

• NEAR 0 = little or no linear relationship

Coefficient of Determination (R²)

proportion of variation in y explained by the fitted regression model

• for simple linear regression with an intercept, use the equation

Residual Standard Deviation (S)

describes the scatter remaining around the line, in the original units of y

• sigma is estimated using the sample standard deviation (s)

• for the case of simple linear regression = estimate of s² is the average squared residue

• it describes the typical size of the differences between observed release times and the times predicted by our line

How do we do Linear Regression?

1. Define the question and variables → what are we predicting and what predictor are we using x

2. Plot the data → look at form, direction, strength, and outliers → can get R without a line

3. Fit the line using least squares → y = bo + b1x

4. Check the assumptions → study design, residual plots, and Q-Q plot

5. Assess the relationship & fit → correlation ®, R², and residual standard deviation (s)

6. Predict and express uncertainty → distinguish interpolation from extrapolation & report an appropriate interval

Interpolation

within our observed range

Extrapolation

outside our observed range