unit 3

what are scatterplots used for

showing relationships between 2 quantitative variables measured for same individuals

describing scatterplots

• direction (pos/neg)

• form (linear vs non-linear; only non-linear if its SUPERRR obvious)

• strength (weak, moderate, strong, perfect)

• outliers and clusters

  • stacked points typically occur in controlled experiments

scatterplot description extra info

• scaling affects how strength and form are perceived

• form can be linear within clusters

explanatory variable

x-var; affects and causes changes in the response variable (explains response)

response variable

y-var; measures outcome of study

choosing x and y for scatterplots

explanatory used for x but can be swapped if end goal is just looking at relationships and not impact of one on the other

association

• positive association: positive slope (high values correspond in both variables)

• negative association: negative slope (high value corresponds to low value)

• write contextual sentence: “as ___ increases/decreases, ____ TENDS to increase/decrease

NEVER EVER EVER FORGET

to use non-deterministic language when writing statements (e.g. tends, on average, predict)

r

• correlation; measures direction and strength of LINEAR relationship between 2 quantitative vars

• value between -1 and 1 with 1 indicating perfect linear relationship and the sign indicating positive or negative association

• non resistant/strongly affected outliers

dont forget the following about r:

• as r gets closer to 0 its a weaker lilnear relationship

• unitless/doesn’t differentiate between explanatory and response variable

• doesn’t describe curved relationships at ALL

• doesn’t summarize data bc diff datasets can have exact same correlation and summary stats

• essentially calculating average product of z-scores

regression

line that describes how response variable changes as explanatory changes

• y hat = a + bx where y-hat is predicted value based on x

written statements for interpretating regression

• equation: y hat is predicted ___ and x is ____

• slope: for every 1 __ increase in [x-unit], WE PREDICT the [y-var] will change by [b]

extrapolation

using regression line for predicting things far outside interval of values of explanatory vars used to obtain line (usually very inaccurate)

strong association doesnt mean

strong extreme predictions

residuals

actual - predicted or y - y hat

• write a sentence to contextualize that the actual is __ more than predicted

• ideally, LSRL has residuals that add up to 0

LSRL (least squares regression line)

• line that makes sum of the squared residuals as small as possible

• calculated w/ linreg(ax+b); MUST be run before calculating residuals

residual plot

• scatterplot of residuals against explanatory variables

• used to figure out appropriateness of linear models; random scatter = linear model good

s

standard deviation/avg size of residuals; typical prediction error

• statement: “when using the LSRL to predict the [y-var] from [x-car], our prediction will typically be off by ___”

r²

coefficient of determination; fraction of variation in y accounted for by the LSRL for y on x

• statement: “About __% of variation in [y-var] is accounted for by a linear model relation [x-var] to [y-var]

• r² can also be calculated by just using r and squaring it

only use graphs

they give you or tell you to make; don’t cite other things

regression output

• explanatory variable is the one under “constant”; look at coeff column

• don’t forget to switch r² value to a decimal and then take the positive and negative square root (determine sign from association or slope)

3 ways to calculate lsrl

• from regression output

• linreg(ax + b)

• standard deviations and means

calculating lsrl from standard deviations and mean

• y = b + mx

• m = r(S_y / S_x)

• y = y bar - m (x bar)

  • where bar indicates the mean

correlation and LSRL are

susceptible to outliers

outliers are influential if

• removing it would change the calculation

  • outliers in x-direction more influential than outliers in y-direction (outliers in y typically just shift line up or down a little instead of rotating the whole line and changing the slope)

• imagine removing it and seeing how it strengthens/weakens/follows the general pattern

how do you determine if the LSRL is a good model

• look at s-value and r² to see how well it fits data

contextualizing LSRL

“of all the lines that summarize relationships between [x-var] and [y-var], the LSRL is the one w/ the smallest sum of squared residuals

dont assume linearity

w/o looking at the scatterplot or residual plot

AND THE IN-GENERAL NEVER FORGET FOR STATS

units :(

correlation doesn’t mean

causation

eyeballing correlation guidelines

0.9 = strong; clear, narrow, linear pattern w/ minimal spread

0.7 = moderate; clear trend obvious but noticeable scatter

0.4 = weak; vague upward/downward tilt but points make wider cloud shape

0 = completely random; diffuse circle or nonlinear curve

• watch for outliers and curves (correlation measures strength of LINEAR relationships)