Correlational analysis

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Scatterplot

Nominal 1. Phi coefficient2. Cramer’s V3. Pearson χ2 test4. Loglinear analysis

Ordinal 1. Spearman’s Rho2. Kendall’s Tau

Interval 1. Pearson’s r

Correlational tests

Correlation coefficient + effect size

Correlation coefficient Shared variance (effect size)

r = 0.0 0.00 Kein Zusammenhang

r = 0.1 0.01 (1%) Geringe Korrelation

r = 0.2 0.04 (4%)

r = 0.3 0.09 (9%) Mittlere Korrelation

r = 0.4 0.16 (16%)

r = 0.5 0.25 (25%)

r = 0.6 0.36 (36%) Hohe Korrelation

r = 0.7 0.49 (49%)

r = 0.8 0.64 (64%) Sehr hohe Korrelation

r = 0.9 0.81 (81%)

r = 1.0 1.00 (100%)

Beispiel: MLU & Age

Child Age in months MLU

123456789101112

242332204358283453464936

2.102.162.251.932.645.631.962.235.193.453.212.84

Long: There is an association between age and MLU. The r of .887 showed that 78.6% (r2) of the variation in MLU was accounted for by the variation in age. The associated probability level of 0.001 showed that such a result is unlikely to have arisen from sampling error.

Short: As can be seen in the table above, there is a strong correlation between age and MLU (r = .887, p = .001).

Beispiel: MLU & Age

Beispiel: Typicality & Frequency

Words Typicality rank Frequency rank

cartrucksports carmotor biketrainbicycleshipboatscat boardspace shuttle

12345678910

12653487910

Kendall’s tau ( = .733, p = .003)

Spearman’s rho (rs = .879, p = .001)

Beispiel: Typicality & Frequency

Phoneme & Silben r = .898, p = .001Phoneme & Häufigkeit r = .795, p = .006Silben & Häufigkeit r = .677, p = .031

Partial correlation

Phoneme & Häufigkeit r = .578, p = .103(Silben Konstant)

Es gibt eine signifikante Korrelation zwischen Geschlecht (Boys vs. Girls) und der Präferenz für ein bestimmtes Spielzeug (mechanisch vs. nicht-mechanisch) (χ2 = 49,09, df = 2, p = .001).

Nominal Daten

Phi-Koeffizient = .70, p < .001Cramer’s V = .70, p < .001

Regression

Correlational analysis gives us a measure that

represents how closely the data points are associated.

Correlation - Regression

Regression analysis measures the effect of the predictor

variable x on the criterion y. – How much does y change

if you change x.

A correlational analysis is purely descriptive, whereas

a regression analysis allows us to make predictions.

Types of regression analysis

Predictor variable

Criterion (target) variable

Linear regression 1 interval 1 interval

Types of regression analysis

Predictor variable

Criterion (target) variable

Linear regression 1 interval 1 interval

Multiple regression 2+ (some of the variables can be categorical)

1 interval

Types of regression analysis

Predictor variable

Criterion (target) variable

Linear regression 1 interval 1 interval

Multiple regression 2+ (some of the variables can be categorical)

1 interval

Logistic regressionDiscriminant analysis

1+1+

1 categorical1 categorical

Line-of-best-fit

20,00 30,00 40,00 50,00 60,00

Age

2,00

3,00

4,00

5,00

6,00

MLU

R-Quadrat linear = 0,786

y = bx + a

Linear Regression

y = variable to be predictedx = given value on the variable xb = value of the slope of the line a = the intercept (or constant), which is the

place where the line-of-best-fit intercepts the y-axis.

Given a score of 20 on the x-axis, a slope of b = 2,

and an interception point of a = 5, what is the

predicted score?

Linear Regression

y = (2 20) + 5 = 45

Beispiel: MLU & Age

Child Age in months

MLU

123456789101112

242332204358283453464936

2.102.162.251.932.645.631.962.235.193.453.212.84

How much does MLU increase with growing age?

There is s strong association between age and MLU (R =

0.887). Specifically, it was found that the children’s MLU

increases by an average of .088 words each months (t =

6,069, p < 0.001), which amounts to about a word a year.

Since the F-value (36,838, df = 1) is highly significant (p <

001), these results are unlikely to have arisen from

sample error.

Linear Regression

Several predictor variables influence the criterion.

Multiple Regression

Plane-pf-best-fit

1. Simultaneous multiple regression

2. Stepwise multiple regression

Eine Universität möchte wissen, welche Faktoren am

besten dazu geeignet sind, den Lernerfolg ihrer

Studenten vorherzusagen. Als Indikator für den

Wissensstand der Studenten gilt die Punktzahl in einer

zentralen Abschlussklausur. Als mögliche Faktoren

werden in Betracht gezogen: (1) Punktzahl beim

Eingangstest, (2) Alter, (3) IQ Test, (4) Punktzahl bei

einem wissenschaftlichen Projekt.

Simultaneous Multiple Regression

Predictor

entrance exam

age

IQ

Scientific project

Simultaneous Multiple Regression

Criterion

final exam

There is s strong association between the predictor

variables and the result of the final exam (Multiple R =

0.875; F = 22,783, df = 4, p = .001 ). Together they account

for 73% of the variation in the exam succes. If we look at

the four predictor variables individually we find that the

result of the entrance exam (B = .576, t = 5.431, p = .001)

and the IQ score (B = .447, t = 4.606, p = .001) make the

strongest contributions (i.e. they are the best predictors).

The predictive value of age (B = .099, t = 5.431, p = .327

and the score on the scientific project is not significant (B =

0.141, t = 1,417, p = 0.168.).

In stepwise regression you begin with one

independent variable and add one by one. The

order of addition is automatically determined by the

effect of the independent variable on the

dependent variable.

Stepwise Multiple Regression

Assumptions Multiple Regression

1. At least 15 cases

2. Interval data

3. Linear relationship between predictor variables

and criterion.

4. No outliers (or delete them)

5. Predictor variables should be independent of

each other

In Linguistics, you often use logistic regression: Multiple

factors determine the choice of linguistic alternates:

1. look up the number - look the number up

2. that-complement clauses - zero-complement clause

3. intial adverbial clause - final adverbial clause

4. aspirate /t/ - unaspirated /t/ - glottal stop - flap

Logistic Regression