ÝSS-20306 Lecture summary
Quantitative
Lecture 1 – Simple Regression Analysis
29-10-2018
Dependence techniques
• 2 different sets of variables
o Outcome
o Predictors
• Example:
o Variable y → Record sales (a.k.a. dependent or criterion)
o Variables xj → attractiveness of band, advertising budget, and number of plays radio
(a.k.a. independent)
o The variable y can be predicted by the three predictors (xj)
• This is used to:
o Predict scores on y on the basis of scores on xj
o To investigate the effect of the xj’s on y
SINGLE PREDICTOR MULTIPLE PREDICTORS
INTERVAL-SCALE PREDICTOR 1. Simple regression 2. Multiple regression
NOMINAL-SCALE PREDICTOR 3. Oneway ANOVA 4. Factorial ANOVA
Interdependence techniques
• To investigate the correlation or association between a number of variables.
• No distinction between outcome and predictor
TWO VARIABLES MULTIPLE VARIABLES
INTERVAL-SCALE PREDICTOR 5. Correlation 6. Exploratory factor analysis
NOMINAL-SCALE PREDICTOR 7. Cross-tabulation 8. Loglinear analysis
Covariance
Statistical model: linear relation
Covariance measures the extent to which positive/negative deviations from the mean on one
variable (proportionally) go together with positive/negative deviations from the mean on the other
variable.
∑𝑖(𝑥𝑖 −𝑢
̅)(𝑦𝑖 −𝑦̅) Covariantie is een parameter die bij
• 𝑐𝑜𝑣(𝑥, 𝑦) = 𝑁−1
= 4.25
twee toevalsvariabelen aangeeft in
• Formula multiplies deviations from means welke mate de beide
• Xi and yi represent the scores on the variables toevalsvariabelen (lineair) met elkaar
• N represents the number of observations samenhangen. (Veronderstelt lineair
verband)
• Means are 5.4 and 11.0
• Terms in numerator are (5 – 5.4)(8 – 11.0) etc.
• To make sense, variables have to be measured on interval-scale
o If this is done, the ratios of the differences between values is meaningful and these
can be used.
• Disadvantage of covariances
o The value that you get, depends on units of the measurement scale (e.g., litres versus
millilitres)
o Not limited to a general, particular range (values can become enormous)
Outcome (y) = red, predictors (xj) = blue
,ÝSS-20306 Lecture summary
Quantitative
Pearson correlation
• Forms a solution to the problem mentioned above, because the values will always fall
between 0 and |1|
• Divides covariance by product of standard deviations → Pearson correlation
𝑐𝑜𝑣(𝑥,𝑦)
o 𝑟(𝑥, 𝑦) = 𝑠𝑥 𝑠𝑦
= 0.87
• Measures linear relationship, so … (at least) interval-scaled variables
o With ordinal data use Spearman’s rho, Kendall’s tau, biserial, point-biserial
• Does not depend on units of the measurement scale
Correlation
Measure of linear relationship
r = 1 assumes perfect linear relation
r = .999 → Positive relation, slope ≈ 1
r = -.999 → Negative relation, slope ≈ -1
r = .763 → Smaller correlation
r = .809 → Not linearly correlated
r = .354 → Increasing ‘mess’
r = .056 → Not even slightly correlated
Statistical inference Fisher Z-transformatie is
• Null hypothesis significance testing (NHST) (are two-tailed!!) een manier om de
o Test H0: r = rhyphotesized versus H1: r ≠ rhypothesized verdeling van Pearson te
1 1+𝑟 veranderen zodat deze
o Fisher z transformation: zr = 2 √𝑁 − 3 ln (1−𝑟) = 1.87
normaal verdeeld wordt.
o Test H0: r = 0 versus H1: r ≠ 0
𝑟 √𝑁−2
o 𝑡𝑟 = = 3.07 N – 2 are degrees of freedom
√1−𝑟 2
• Assumptions that need to be met
o Independent observations
o Variables normally distributed (to make sure that the p-value is correct)
o Assumptions necessary for applicability of theoretical distributions, i.e. validity of p-
value
o Sample obtained by simple random sampling (all have the same chance to enter the sample)
• Also possible to create (e.g. 95%) confidence intervals
o If we draw same-sized samples over and over again, 95% of the correlations will be in
this interval
Outcome (y) = red, predictors (xj) = blue
,ÝSS-20306 Lecture summary
Quantitative
Measure of relationship
Correlation
• Effect size r2
• Field (citing Cohen, 1988, 1992) Others (citing Cohen, 1988, 1992)
0.01 is small 0.01 is small
0.09 is medium 0.06 is medium
0.25 is large 0.14 is large
• Also called coefficient of determination (proportion of variance accounted for)
Dependence
Simple regression
• Goal and diagram
• Predict outcome variable (criterion/dependent) y from predictor variable x (independent)
• Investigate effect of x on y
Adverts Packets
watched bought
Simple regression Interesting
Model Typically uninteresting
• Regression equation Error/residual
yi = f(xi) = (b0 + b1x1) + εi
= model + error/residual
Regression weights/coefficients
B0 (intercept)
B1 (slope)
Estimated such that variance (εi) is as small as possible
(method of least squares) →
Estimates: b̂0 and b̂1
Predicted scores: ŷi = b̂0 + b̂1xi
𝑐𝑜𝑣(𝑥,𝑦)
• b̂1 will be 𝑣𝑎𝑟(𝑥)
o Best (smallest variance)
Linear
Unbiased (expectation is equal to true b1 in the population)
Estimator (BLUE)
o If (assumption) εi = independent, identically distributed N(0,σ)
• Based on measure of linear relationship, so … assumes, (at least) interval-scaled outcome
and predictor variables
• Assumption of normally distributed residuals also requires (at least) interval-scaled outcome
variables
• Overall statistics: r(y, ŷ) = R = (multiple) correlation coefficient
• R2 = coefficient of determination
Outcome (y) = red, predictors (xj) = blue
, ÝSS-20306 Lecture summary
Quantitative
Overall statistics & SPSS
Test H0: R = 0 versus H1: R ≠ 0
Detailed statistics
Extra: t-test by simple regression: dummy variable
T-test: test of the average in the first group differs from the average from the second group
Assumptions
• Homogene variances (variance in the first group is about as big as variance in the second group)
• The larger the sample, the smaller the significant coefficients will be (so, pay attention to scale)
Outcome (y) = red, predictors (xj) = blue
