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 sizeCorrelation coefficient Shared variance (effect

size)r = 0.0 0.00 Kein Zusammenhang

r = 0.1 0.01 (1%) Geringe Korrelationr = 0.2 0.04 (4%)r = 0.3 0.09 (9%) Mittlere Korrelationr = 0.4 0.16 (16%)r = 0.5 0.25 (25%)r = 0.6 0.36 (36%) Hohe Korrelationr = 0.7 0.49 (49%)r = 0.8 0.64 (64%) Sehr hohe Korrelationr = 0.9 0.81 (81%)r = 1.0 1.00 (100%)

Beispiel: MLU & Age

Child Age in months MLU123456789101112

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 rankcartrucksports 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 intervalMultiple 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 regression2. 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

Predictorentrance examageIQScientific project

Simultaneous Multiple Regression

Criterionfinal 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