POLI 380 - Midterm 2

Standard deviation (SD)

The average distance of the observations to the mean.

Standard deviation function (R)

sd(X , na.rm = T)

Standard deviation formula (R)

sd(X) = sqrt(sum(obsevation_x - mean_x)² / sum(x))

Normal distribution

Range of data where 95% of the observations fall within 1.96 SDs from the mean.

Z-scores

Distance from the mean in standard deviations for a individual observation.

Z-score formula (R)

observation_x - mean_x / sd(x)

Centering

Center the data around the mean to determine if the observation is above or below the mean.

Scaling

Adjusts the data to to tell us how many SDs away an observation is from the mean.

Variance

Square of the standard deviation, used to measure spread of distribution.

Scatterplot

Visualize relationship between two variables by plotting X against Y.

Correlation Coefficient

Summarizes the direction and strength of the linear association between the two variables.

Correlation Coefficient Formula (descriptive)

Sum of the product of the z-scores of X and Y divided by the number of observations.

Correlation Coefficient Interpretation: Line of Best Fit

cor > 0 = line of best fit is positive

cor < 0 = line of best fit is negative

Correlation Coefficient Interpretation: Strength of Linear Association

|cor| = 0 : no linear association

|cor| = 1 : perfect linear association

Process of Prediction

Fitting a model and making predictions.

Predictors

Variables that are used as the basis for our prediction (X).

Outcome Variable

Variables that are used to measure what we want to predict (Y)

Y(hat)

The predicted value of Y based on the values of X and the model that summarizes the relationship between X and Y.

E(hat) = Y - Y(hat)

The errors of our predictors.

Line coefficients

Intercept & Slope

Intercept - a(hat)

Specifies the vertical location of the line.

Slope - b(hat)

Angle or steepness of the line.

Line formula.

Y(hat) = a(hat) + b(hat)X

lm()

Used to fit a linear model. lm(Y ~ X, data = df)

Fitted line

Used to measure the linear relationship between X and Y.

Fitted Line Formula

Y(Hat) = a(hat) + b(hat)X

Residual

The difference between the predicted value and the actual value.

Least squares method

Used to minimize residuals by finding a fitted line where the sum of the square of the prediction errors is lowest.

Model Fit

How good does the model do in making predictions?

R²

Measure the proportion of the variation in the outcome variable explained by the model.

Impact of R²

Higher value means that the model better fits the data.

Formula of R²

R^2 = cor(X , Y )^2

Linear Models - Best Practice

Find X variables that are correlated with Y.

Observational Data

Data collected about naturally occurring events,

Confounders

Variables that affects both the treatment and outcome variables.

Issue of Confounders

Confounders obscure the causal relationship between X and Y.

Relationship between slope and difference-in-means

Same value.

Relationship between intercept and control group.

Same value.

Multiple Linear Regression

Linear regression with more than one treatment variables.

Interpreting Coefficients in Multiple Linear Regression

Intercept is Y when all X are 0. (A)

Slope is the change of Yj associated with the change of Xj = 1 while holding all other X variables constant. (B)

Confounding and Muliple Linear

Adding confounders as controls in the model makes treatment and control group s comparable after controls.

Controlling Confounding

Adding Z as a control variable in the model makes the values of Z constant.

Probability

Proportion of specific events over infinitely many identical trials.

Probability Distribution

Likelihood of each value can take.

Sample Variation

Variance of the sample’s observation.

Effects of Sample Size

Larger sample sizes are closer to the true effect of the population.