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univariate data
one variable data set
bivariate data
relationship between two variables
explanatory variable
predict or explain changes in response varaible
response varaible
measures an outcome of a study
scatterplots
- shows relationship/association between two quantitative variables measured on the same individuals
- explanatory variable: x-axis
- response variable: y-axis
- no explanatory variable: either variable can go on x-axis
describing scatterplots
- direction: positive, negative, no association
- form: linear or nonlinear
- strength: weak, moderate, strong
- unusual features: points that fall outside of overall pattern and distinct clusters of points
correlation r
- measures the direction and strength of association for a linear relationship only
- between -1 and 1
- does not equal causation
- does not measure form
- not a resistant measure of strength
- both quantitative variables
- no distinction between explanatory and response variables
- does not change when units change
- no unit of measurement (just a number)
correlation r interpretation
"the linear relationship between X and Y is STRENGTH and DIRECTION"
coefficient of determination r² interpretation
"the percent of the variation in Y explained by the linear relationship with X"
used to make predictions
ŷ = a + bx
residual
actual - predicted
(difference between the actual value of y and the value of y predicted by the regression line)
residual interpretation
"the actual CONTEXT was RESIDUAL above/below the predicted value for X = #"
ŷ = a + bx interpretations
- "when X = 0 CONTEXT the predicted Y-CONTEXT is Y-INTERCEPT"
- "for each additional X-CONTEXT the predicted Y-CONTEXT increases/decreases by SLOPE"
regression line
summarizes relationship between two variables but only when one variable helps explain the other
extrapolation
- using a regression line to make a prediction for x-values outside (larger/smaller) the x-values used to obtain the data
- don't do it; not accurate
least-squares regression line
the line that makes the sum of the squared residuals as small as possible
residual plot
- scatterplot that displays the residuals on the vertical axis and the explanatory variable on the horizontal axis
- appropriate model: no leftover curved pattern
- not appropriate model: leftover curved pattern
correlation r strength
- strong negative: -1
- moderate negative: -0.5
- weak (no association): 0
- moderate positive: 0.5
- strong positive: 1
properties of correlation r
- unusual value in pattern = strengthens r
- unusual value not in pattern = weakens r
standard deviation of the residuals s
- measures the size of a typical residual
- s measures the typical distance between the actual y values and the predicted y values
coefficient of determination r²
- measures the percent of variability in the response variable that is accounted for by the LSRL
- tells us how much better the LSRL does at predicting values of y than simply guessing the mean y for each value in the data
regression to the mean
for an increase of 1 standard deviation in the value of the explanatory variable x, the LSRL predicts an increase of r standard deviations in the response variable y
high leverage in regression
much larger or smaller x-values than the other points in the data set
outlier in regression
- does not follow the pattern of the data
- large residual
influential point in regression
if removed, big changes to slope, y-intercept, and r values
association does not imply causation
a strong association is not enough to draw conclusions about cause and effect
horizontal outliers
tilt line
vertical outliers
shift line up/down
linear
graph x vs. y
exponential
graph x vs. log y
power (y=axᵖ)
graph log x vs. log y
achieve linearity with power model
- raise value of explanatory variable x to the p power (xᵖ, y)
- take pᵗʰ root of the values of the response variable y (x, ᵖ√y)
linear pattern
scatterplot of logarithms of both variables
roughly linear assoication
scatterplot of logarithm of y against x
choosing the best regression
1. check scatterplot for linear pattern
2. check residual plot for no distinct pattern
3. check for the r² that is closest to 1
