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Correlational Research Methods - Summary, Tilburg University

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Correlational Research Methods

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Tilburg University (UVT)

A summary of the course Correlational Research Methods. The summary consists of the lectures given. If you have any questions, you can message me :)

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psychologie
psychology
tilburg
samenvatting
summary
correlational research methods

Institution
Tilburg University (UVT)
Education
Psychologie
Course
Correlational Research Methods

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Lecture 1
Samples vs. Population
Types of Sampling Designs
 Simple random sampling = every member in the population has an equal chance to be
sampled.
 Stratified sampling = the population is divided into strata; within each stratum a random
sample is drawn.
 Convenience sampling sample of people who are readily available.

Descriptive vs. Inferential Statistics
 Descriptive statistics
o Summarizing data using measures of central tendency (mean, median, mode) or
measures of dispersion (variance, standard deviation).
 Measures about central tendency and dispersion are only about one variable
(correlation is about two).
 Inferential statistics = making generalization about the population.
o Procedures:
 Null hypothesis significance testing
 Steps:
1. Formulating H0 and H1.
2. Make a decision rule.
3. Obtain a t- and p-value from the output.
4. Either reject or keep H0 and draw a conclusion.
 Reject if it is in critical regions or p-value < .05
 Accept if it is in ‘normal’ region or p -value > .05
 Confidence interval estimation
 Definition: when we carry out an experiment over and over again,
the 95% CI will contain the real value of the parameter of interest in
95% of the cases.
 Interpretation: based on the data, this would be the most probable
range of values for the real value of the correlation coefficient.
 Importance: gives an indication of how precise the point estimate is.
 Using CI for hypothesis testing: look whether the value of the
statistic under H0 (usually 0) lies within the interval.
o If it is, H0 will belong to the values from which we have 95%
certainty that it could possibly be the population value.

Level of Measurement
 Classical measurement levels: nominal, ordinal, interval, ratio.
 Categorial variable (e.g., gender).
 Quantitative variable (e.g., IQ).
o More continuously measured.

Experimental, Quasi-Experimental, Correlational Studies
Correlational Studies
 To measure the relationship between variables.
 Pearson’s correlation coefficient.
o For measures of linear association between variables.
o Notation:
 ρ = correlation in the population.
 r = correlation in the sample.

, o -1 ≤ r ≤ 1
o r = 0: there is no linear association (but there might be non-linear association).
 Statistical tests for the correlation coefficient:
o Inferential statistics:
1. H0: ρ = 0 vs. H1: ρ ≠ 0

 T-test: t=r
√ N −2
1−r 2
with df = N – 2

 SPSS gives the (two-sided) p-value for this test.
2. H0: ρ = c vs. H1: ρ ≠ c
 C is a number between -1 and 1, but not 0.
 Fisher Z-transformation and Z-tests are required (to test whether the
correlation in the sample is significantly larger than e.g., 0.8).
 Not available in SPSS.
 P-value = the p-value is the probability of the data in the sample (r) or more extreme (further
away from ), given that H0 (p = 0) is true.
o Use:
o Decide which significance level to use (usually 5% or 𝛼 = 0.05).
o When p < 𝛼, reject H0.

Lecture 2
Confidence Intervals for r:
CI ( 1−a) 100% =r ±Crit . Value(a ,two tailed ) × SE(r )
 Crit. Value = the critical value depends on the desired confidence level.
 SE(r) = the standard error.
o Describes variability in values of the sample (r) if you draw a large amount of samples
from the population.
 CIs for correlation coefficients are not symmetric.
o Meaning r is usually not in the middle of the CI (due to Fisher transformations)).
 Smaller N = wider interval – less precise.
 Large sample = smaller CI – more precise measurements.
 A 90% interval would be narrower than a 95% interval – gives more precise estimation.
 Less certainty means more accuracy.

Assumptions for r:
 Independence among observations.
o Is satisfied when a random sample has been drawn.
 X and Y scores have a bivariate normal distribution.
o Scatterplot shaped like cigar.
 X and Y are linearly related.
 Assumption of homoscedasticity: scores on Y should (roughly) equal variance across levels of
X.

Power and Multiple Comparisons
 Power = the probability to reject H0 given that H1 is true (there is really an effect).
 Larger N  smaller CI and more power.
 To find small effects (p is small), a larger N is needed.
o Carry out power analysis before gathering data.
o N > 100 (to check assumptions, less impact of outliers).
 When multiple comparisons are made (multiple correlations are tested), the probability of
making a Type I error (= incorrectly rejecting H0) will increase.

, o Replication
o Cross-validation.
o Bonferroni correction PC a =EW a /k

Squared Correlation
 Another way to report effect size.
2
 r XY = the proportion of the variance X you can linearly predict from Y – and vice versa.
o E.g., number of hours studying and exam grade correlate 0,40; thus 0.4 2 = 0.16
(16%) of difference in exam grades can be predicted by differences in the
number of hours studied.

Simple Linear Regression Analysis
 One independent variable X, and one dependent variable Y.
 XY
 Linear relationship means we can predict Y from X using a
'
linear function: Y =b 0+ b1 X
o Y ' = the predicted value of Y given X.
o b0 is the intercept; the predicted value Y ' when
someone scores 0 on X.
 In practice often not very interesting.
o b1 is the regression coefficient; the change in Y ' when X increases with one unit – the
slope of the line.
o b0 and b1 are called parameters but b0 is not a predictor.
 Simple regression analysis:
1. Find the best fitting straight line – find values for the coefficient (b0 and b1) – for
which we can best predict Y from X.
 Line for which prediction errors are smallest (e i)
 Choose b0 and b1 as such that e i is as small as possible.
 This can be done by using ‘least squares estimation’:
N
 ∑ ¿¿
i=1
 H0: b1 = 0 vs. H1: b1 ≠ 0
 Least squares estimators for b0 and b1 can also be calculated from r XY and
the standard deviations (sx and sY).
sY
 b 1=r
sX
X−X Y −Y
 r =∑ ( Z ¿ ¿ X × ZY )÷ N , where Z X = ∧Z Y = ¿
SX SY
 SPSS DOES THIS USUALLY.
 Regression line always goes through the point where the averages intersect.
 Estimated regression model:

, 2. Decide how well you can predict Y: inspect individual prediction errors.
 e i ¿ Y i −Y 'i
 e i is the prediction error for each person i (i = 1, i =2, …, N).
 Difference between observed and predicted value Y.
 The sum of the prediction errors will equal 0 – with a little deviation because
of rounding up or down.
 Thus, the average prediction error is also 0.
 Total variance = predicted variance + error variance:
 s2Y =s2Y + s 2e
'

 The variance of the prediction error is equal to the unexplained variance.
 Process:
o Calculate 1-R2, which the proportion of unexplained
variance.
o To calculate total unexplained variance: proportion
2
unexplained variance times sY
 R2 is the proportion explained by the variance.
 1−R 2Y × X is the proportion of unexplained variance.
 Ways to calculate R2
o
explained variance∈Y based ont h e regression SSregression
=
total variance∈Y SStotal
o With ONE predictor in model: R = absolute value of 𝛽1, so
squared 𝛽1 = R2
s 2Y '
o 2
s Y
3. Check whether you can generalize the results to the population level.
 To answer this question, we use hypothesis tests:
 H0: b1 = 0 – there is no linear association in the population.
 H1: b1 ≠ 0 – there is a linear association in the population.
 Test statistics:
b^ 1−b1
o t= : t-distribution, with df = N -2
SE ( b^ 1 )
o The SE( b^ ¿ ¿1) isthe standard error of b^ 1 ¿
o Note: df = N - #predictors -1 = dferrors’

Lecture 3
Interpretation of the Estimated Regression Coefficient b1
 Y ' =b 0+ b1 X
 First: two ways to interpret Y ' .
o Y ' = predicted Y given someone’s score on X.
o Y ' is an estimation of the average score on Y
for the population of people with a certain
value on X.
 Interpretation regression coefficient b^ 1:
o When X increases with 1 unit, then Y ' increases with b^ 1 units.

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