10. Correlation and Regression Flashcards
what is the bivariate case?
the case where there is one predictor (IV) and one criterion variable (DV)
what is the abbreviation for pearson’s correlation coefficient?
r
what is the range of pearson’s correlation coefficient?
-1 and +1
what do negative and positive values of pearson’s correlation coefficient indicate??
the direction and strength of a linear relationship between an X and Y variable
what does a negative pearson’s correlation indicate?
indicates a relationship where increases in one variable are associated with decreases in the other and vice versa
what does a positive pearson’s correlation indicate?
indicates a relationship where increases in one variable are associated with increases in another and decreases with decreases
what is a pearson’s correlation coefficient designed to test?
for a linear relationship
what is pearson’s correlation coefficient based on?
the ability to invisibly draw a straight line through the data point
what must a pearson’s correlation coefficient be?
a linear relationship
what is the formula for pearson’s correlation coefficient (r)?
r =
N
what is the procedure for calculating r?
for each score, calculate its corresponding Z score (Z_x and Z_y) ensuring that you use the correct mean and Sd
Multiply the Z_x by the Z_y to get the cross product ensuring that the sign (+ or -) is correct
add all the crossproducts (ΣZ_XZ_Y)
and divide by the number of pairs of scores
what do Z scores tell us
whether a score (X or Y) is above (+Z) or below (-Z) a mean (M)
what do we get if we multiply Z values on X and Y for each person?
crossproducts
what does it mean when we get mostly positive cross products?
positive correlation
what does it mean when we get mostly negative cross products?
negative correlatin
what does it mean when we get an equal number of + and - cross products?
no correlation
what are the characteristics of a positive correlation?
- mean on X (+ZX), above mean on Y (+ZY)
- Below mean on X (-ZX), below mean on Y (-ZY)
- ZXZY values are mostly positive (multiply two negatives = positive)
what are the characteristics of a negative correlation?
- Above mean on X (+ZX), below mean on Y (-ZY)
- Below mean on X (-ZX), above mean on Y (+ZY)
- ZXZY values are mostly negative
what are the characteristics of no correlation?
- Position on X (+ZX or -ZX) not linked to position on Y (+ZY or -ZY)
- ZXZY values are equally positive and negative
what does it mean if r is closer to +1
there is a strong positive relationship
what does it mean if r is close to -1
strong negative relationship
what does it mean if r is close to 0
weak relationship
what can correlation tell us?
the direction of our relationship
what can regression help us with?
helps us actually plot the line that the correlation metaphorically draws through the data
what can the line that regression assists in drawing help with?
helps us to predict scores on our DV
what is correlation coefficient a measure of
how close to the line the data is
why is it called the line of best fit?
because it is drawn in the position that minimises the distances of alll the data points
what does the line of best fit influenced by??
every data point in the scatterplot
where does the line of best fit carefully place itself?
in a position that overall is closest to all teh data points that it can be
what are we calculating with regression
the ordinary leased regression (OLR)
what are the two pieces of information that we need to calculate the line of best fit based on our x and Y scores?
the slope and the Y-axis intercept
what is the slope?
how steep it is
what is the Y-axis intercept?
where the line starts from on the y axis
where the line of best fit intercepts the y axis
it is the predicted value of Y when X is zero
what is a slope?
an indication of the gradient of the line OR its steepness
what is the slope also known as?
the regression coefficient
what is the slope in mathematical terms
it is how many units of the Y variable (DV) you increase for every unit increase in your X variable (IV)
what is the formlua for the slope?
b =
S_x
what does b stand for?
the gradient
what does a larger b figure mean?
a stronger gradient between X and Y - the steeper the line
what is the gradient another term for?
steepeness or the slope
instead of Z scores to represent movement on our variables, wht are we using?
deviation scores
what is the equation for the sum of squared deviations for the X variable?
SS_x =
∑ (X - M_x)^2
what dont we need to do for the exam?
dont need to knwo how to calculate the linear regression equation from scratch but you do need to interpret regression on an SPSS output
what is the letter that represents intercept?
a
how do we find the y interept?
multiply our slope (b) by the mean of our IV (X) and then subtract that from the mean of our DV (Y)
what is the equation for y intercept?
a =
M_y - (b)(M_x)
what is the regression equaiton?
Ÿ = a + (b)(X)
what does the regression formula allow us to do
allows us to predict what a person;s score on the Y variable is likely to be if you know their score in the X variable and the relationship between the X and Y variable
what is Ÿ?
the predicted score on the Y variable
what does the hat ontop of the y indicate?
it is “predicted” as opposed to an actual score
what is a?
the y-intercept. or the point at which the regression line crosses the Y axis (hence where x equals 0)
what is b?
the slope. the gradient or steepness of the regression line
what is X?
the person’s score on the X variab;e
what is a residual score?
the difference between the actual score and the predicted score
what is the equation for residual?
R_y = Y - Ÿ
what does a residual score indicate
how far away from the predicted score the actual or raw score is
what is the standard error of estimate (SEE)
the average or mean of the residual scores would represent for us the average distance of our data points to the regression line
it also represents the average that we would be in error if we use the regression line to predict a persons score
what would a high value of the SEE indicate
that on average the prediction is inaccurate
low SEE means…
good for making predictions
what unit is SEE always in the unit of?
the unit that the DV is measured in
what do we get if we sum all the residuals?
0
what is the formula for SEE?
SEE =
Square root of:
N
what is the SEE an underestimate of?
the true population SEE and so a correction is to be made
what is the correction made when calculating the estimate population of SEE
instead of N you use N-2
what is the equation for the population SEE?
SEE =
Square root of:
N-2
what are the two possibile partitions of the variability in Y-scores
variability due to error (SS_error)
varaibility due to regression (SS_reg)
what will be the total variability (SS_total)?
SS_Y
what is SS_y
the sum of squared deviations between each Y score and the mean of Y
what will be the SS_error?
the sum of squared deviations between each Y score and the predicted Y score
what is the regression variability (SS_reg)
the sum of squared deviations betwene each predicted Y value and the mean of Y
what df do we use to calculate the MS with regard to these SSs??
1 and N-1
what is the f-statistic for regression
MS_reg / MS_error
what does the f-stat for regression as>
is our model significantly predicting?
what is r^2 within an ANOVA context equivalent to?
SS_regression / SS_total
what is an r^2
similar to eta squared in ANOVA
when F observed is greater than F critical what does this indicate?
that regression is not significant
what are the assumptions for correlation and regression?
normality and linearity
what is the assumption of normality?
regression and correlation assume that both the X and Y variables have relatively normal distributions
what is the assumption of linearity
peason’s correlation and linear regression both assume that the relationship they are testing is linear
what is the potential problem for correlation and regression
impact of outliers / extreme scores
what is an outlier
data point that lies away from the rest of the pack of data
what does correlation not imply?
causality
