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Samenvatting Construction and analysis of questionnaires

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Tilburg University (UVT)

What is covered? - Basic of statistics - the one-sample T-test - the independent sample T-test - one-way-between-subjects analysis of variance (ANOVA) - Bivariate pearson correlation - Bivariate regression - Adding a third variable - Multiple regression analysis with 2 predictors - Du...

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CAT – Summary
The basics
Deviation scores: the difference between 1 score and the mean score: X – M
 Sum of squares (SS): (X-M)². You need this to be able to calculate sample variance and
standard deviation.
Variability: how large are the differences between scores.
Degrees of freedom (DF): N-1, is needed to calculate S² and SD.
Sample variance (S²) = SS/(N-1)
Standard deviation (S/SD): √S² or √SS/DF. Standard deviation tells us how dispersed the data from
the mean is.
Normal distribution: a fixed relationship between distance from the mean and area under the curve.
Z-score: index of the distance of an x-score from the sample mean that has been converted into unit
free or standardized scores. It tells us how extreme or normal a certain score is.
Z = (X-M)/SD. Z-score is 1.96 at a two tailed alpha 0.025.

The one-sample T-test
Null-hypothesis significance testing (NHST/H0): uses sample mean and SEm to answer questions
about hypothesized values of the population mean.
Alternative hypothesis can be:  , >, < between variables.  is two-tailed and the others one-tailed.

Assumptions one-sample T-test
 Does not use normal distribution
 (M- hypothesized mean)/SEm is used
 Uses tables for T-ratio
o Large T-rate tells you that your obtained value of M is unusual, and that you can
reject the null-hypothesis.

You do not use one-sample T-test multiple times in your sample population, because your change of
finding a significant result (an exceptional result, that does not match H0) becomes larger with every
try.
 Type I error occurs: rejecting the H0 while it is correct.

Cohen’s D: gives an index of the effect size. So, how much has the result of the sample-T-test effect.
The value of T-test depends on Cohen’s D (effect size) and N (sample size). If T is significant then you
reject the null-hypothesis.

The independent-sample T-test
Homogeneity assumption: means that variation in the populations are equal – equal variance is
assumed. How do you know if this assumption is violated?
 Look at the Levene’s test in SPSS: the test needs to be not significant, otherwise this
assumption is violated.
Eta Squared (N²): an estimate of the proportion of variance in 1 or more dependent variable scores.
N² = T²/T² + DF

,One-way-between-subjects analysis of variance (ANOVA)
ANOVA: Is used to analyse one or more variables. It looks if these group averages differ from the
population average.
 It cannot tell which group (sample) differs, but only that it does or does not.
 You can look at the variance within groups, and the variance between groups.
 F-distribution is used.

Assumption of ANOVA:
 Quantitative dependent variable of interval/ratio (continuous) measurement level.
Independent variable of nominal measurement level.
 In the full sample and in each group, scores of the dependent variable are approximately
normally distributed.
 No outliers
 Variances of scores of dependent variables is equal between groups. Homogeneity is
assumed.
 Observations have been selected by random sampling and are independent.

Yij = individual score
My = population mean
Mi = group mean
αi = difference between group average and population mean

SSbetween = (MI – MY)² SSwithin = (YIJ – MI)²

F-ratio test statistic = MS between/MSwithin
MSbetween = SSbetween/(k-1)
MSwithin = SSwithin/DF1
K = number of groups

F-critical value = look up in table
 If the F-ration larger is than the critical value then you reject the null hypothesis
 F-critical value determines when a number is significant in the sample population

Levene’s test: if the test is significant then that means that there is not equal population variance.
Which means that you have violated the homogeneity assumption. Which means that you cannot
use the F-test. If it is not significant you can look at the F-test and P-value, if this is significant than
the means of the groups vary, if it is not significant you accept the Null-hypothesis. You have to look
at the Welch-test or Brown-Forsythe test, to know if you ANOVA test is significant.

Effect of the group: Ak = Mk (mean of random score)– My (total mean)

, Bivariate Pearson correlation <-->
Assumptions of Pearson’s correlation
 Variables are quantitative or both dichotomous
 Variables are linearly related
 Variables have a bivariate normal distribution
 No extreme outliers  otherwise value of R can be inflated or deflated by outliers.
 Homoscedasticity: Y-scores have the same variance across level of X, and vice versa.
Heteroscedasticity is varying variance
Pearson’s R is used to describe the strength of a linear relationship. It is always 1 or -1
 Standardized
 Always between 2 continuous/dichotomous variables
 If x increases, than Y increases.
 When there is no linear association, that does not mean that there can’t be another kind of
association

Computation of Pearson’s R:
1. Convert scores X on Y to Z-scores (unit free)
a. Zx = (X – MX)/Sx
b. Zy = (Y-MY)/Sy
2. Multiply Zx X Zy for each score
3. Sum of each score = Zx x Zy
4. Zx x Zy /N-1 = R

Testing H0:P0 = 0  no correlation between X and Y
P: population value of the correlation
Ratio fort = T = R-P0/SEr SEr = √1-r²/ √N-2

How to limit type 1 error in correlations:
 Limit the number of correlations
 Cross-validation
 Bonferroni (a/k (significance test). This lowers the significance level
 Replicate correlations in new samples.
Factors that influence magnitude of R:
 Pearson’s R will be deflated if X and Y are non-linear or curvilinear
 Pearson’s R can be used as a standardized regression coefficient to predict the standard
score on Y from the standard score on X, vice versa. Z’y = Rx x Zx  this says that if X is 1
standard deviation from its mean, then Y will be R-standard deviations units from the mean.

A perfect R-score (so, 1) can only occur when X and Y have the same SD (distribution shape) this does
not occur a lot. When it does than R cannot be used as a regression coefficient. The mean, variance
and distribution shape of scores should be similar to the mean of the population.
 Samples that include members of different groups can provide misleading or confusing
information
 If X has poor measurement reliability, it’s correlations with other variables will be attenuated
(Reduced)  magnitude of attenuation: Rxy = Pxy √Rxx x Ryy. Rxy is observed correlation
between x and y. Pxy is real correlation (Without errors). Rxx x ryy is reliability of the
variables.
 Do not have the same survey questions

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