Statistics!!!

Predictor variable

A factor used to predict changes in a dependent variable

Outcome variable

What researchers aim to study the change of

Prediction residual (error)

actual value - predicted value

Prediction

the value your model thinks y will have for a given x

Random process

a collection of variables that change over time, but in a way you can’t predict exactly, just probabilistically

Random variable

categorical outcomes of random processes translated into numerical representations

Random event

Something that might happen, but you can’t know for sure until it actually happens

The goal of prediction

To find a rule (model) that makes the smallest possible prediction errors overall

Choosing predictor and outcome

Outcome = Y, Predictor = X

Basic linear regression equation

Y(hat) = a + bX. a = intercept = predicted Y when X equals 0. bX = slope = change in predicted Y for each 1 unit increase in X

Least squares

Chooses a and bX so that the sum of squared residuals is as small as possible

General linear model

Doesn’t require the optimization of the smallest sum of squared residuals like “least squares”

R gives:

Intercept = 2.5

Slope = 0.8

Y(hat) = 2.5 + 0.8x

Y (hat)

Predicted value

Interpret slope b (numerical X)

For each 1 unit increase in X, predicted Y increases by b units

Interpret intercept a (numerical X)

Predicted value of Y when X = 0

Interpret slope b (categorical X with 2 levels)

If X = 0(group A) or 1 (group B), then:

b = difference in group means. (How much higher/lower B is compared to A)

Interpret a (Categorical X)

a = predicted outcomes for the baseline group (group coded 0)

What is R²

The proportion of variation in Y explained by the model. Example: R^{2 = 0.40 means 40% of differences in Y are explained by X}

Finding R² from r²

R² = r²

How to find r from R²

r = the square root of R²

Predicting using a linear model

Just plug X into Y(hat) = b0 + b1x

Calculating residual

Residual = Y - Y(hat)

4 conditions for least squares regression

Linear relationship. Independent observations. Normal residuals. Constant variance (spread of residuals stays roughly the same across X) LINC

Normal residual

The models errors are roughly bell shaped, centered at zero, and not skewed

Checking for the 4 conditions in a residual plot

Look for linearity, constant variance

Checking for the 4 conditions in a histogram/QQ plot of residuals

symmetry, unimodality, most residuals close to 0, few large residuals (light tails), just be bell-shaped

Checking study design for the 4 conditions (check independence)

Each data point comes from a different individual or independent event. One person’s measurement doesn’t affect another’s. No pairing, no matching, no repeated measures. Data weren’t collected in a way that clusters people

What is unreasonable extrapolation?

Predicting for X-values far outside the range of observed data, which may give nonsense results

High leverage

Unusual X-value

Influential

Point that changes the regression line a lot

Random process

A repeatable process with uncertain outcomes

Outcome

One result of the random process

Event

A collection of outcomes

Disjoint probability

P(A and B) = 0

A and B are independent if

P(A|B) = P(A) or P(A and B) = P(A)P(B)

Addition rule (disjoint A, B)

P(A or B) = P(A) + P(B)

Multiplication rule (general)

P(A and B) = P(A|B)P(B)

Multiplication rule (independent A and B)

P(A and B) = P(A)P(B)

Using tree diagrams (joint probabilities)

Multiply along branches

Using tree diagrams (total probabilities)

Add branches

Using probability tables (marginal probability)

Row/column totals

Using probability tables (conditional)

Cell/row or column

Using probability tables (Joint)

Individual cell value

Probability distribution

A list of all possible values of a random variable and their probabilities

Expected value (mean)

E(X) = win-probability(value) - loss-probability(value)

Interpretation: long-run average value of the random variable.

Variance

Step 1. Find the mean. Step 2. Subtract each individual value from the mean. Step 3. Square all the new values. Step 4. Sum all squared values together and divide by n - 1 if sample population. Divide by n if whole population.

Standard deviation

Square root of variance

Interpretation: average distance from the mean.

Linear combinations (aX + b)

• Expectation:
  E(aX+b) = aE(X)+b

• Variance:
  Var(aX+b) = a^2 Var(X)

Sums of variables (X + Y) if independent

Mean: add the means

Variance: add the variances
(Never add SDs)

Var(X) =

E[X - E(X)to the second power] Or

Var(X) = E(X)to the second power - (E(X)) to the second power

E(aX + bY)

aE(X) + bE(Y)