Amsterdam Rehabilitation Research Center | Reade Correlation and linear regression analysis Martin van der Esch, Phd

Amsterdam Rehabilitation Research Center | Reade

Correlation and linear regression analysis

Martin van der Esch, Phd


Content

Correlation and linear regression analysis

Association researchHowever, also used in experimental studies


Correlation and regression

- Interested in relationship/association/correlation- Direction and magnitude of relationship- Dependent or independent variables- Association does not imply a ‘cause and effect’ relationship


Correlation


Correlation

Expressed as productmomentcorrelation Pearson coefficent (r) when data are not skewed

or rank order correlation Spearman (rs) when data are ordinal, skewed or in case of presence of outliers.

DimensionlessRage between +1 and –1 (0 = no correlation)Magnitude indicates how close the points are to a

straight line (the strength of an association)

+1 or –1: perfect correlation: all points lying on the line

5


Between -1 to 1.


70605040302010

Leeftijd

220

200

180

160

140

120

100

Sy

st.

blo

ed

dru

k

R Sq Linear = 0,432


70605040302010

Leeftijd

220

200

180

160

140

120

100

Sy

st.

blo

ed

dru

k

R Sq Linear = 0,712


Correlation coefficient

Range: -1 ≤ r ≤ 1.

In SPSS Model Summary

,844a ,712 ,702 9,563Model1

R R SquareAdjustedR Square

Std. Error ofthe Estimate

Predictors: (Constant), Leeftijda.

Coefficientsa

97,077 5,528 17,562 ,000,949 ,116 ,844 8,174 ,000

(Constant)Leeftijd

Model1

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: Syst. bloeddruka.


Formula correlation

n

i

n

iii

n

iii

yyxx

yyxxr

1 1

22

1

)()(

))((


Regression analysis


Statistical analysisData were analyzed with SPSS for Windows 16.0 (SPSS Inc). According to their distribution, the

various parameters are expressed as mean (± standard deviation) or median (interquartile range). Data with a non-Gaussian distribution was log transformed for analysis if possible. To compare the groups, student’s T-test or Mann-Whitney U test was used when appropriate. Furthermore, correlations between variables were analyzed by using Pearson correlation or Spearman’s rho tests. Univariate linear regression analyses were performed on log-transformed data to investigate the influence of possible confounders (i.e. sex, smoking status, systolic blood pressure and body mass index (BMI) on the results. Wilcoxon signed-rank test was used to investigate the differences in values at baseline and at 8 weeks in the prospectively followed subgroup of patients (n=9). P-values less than 0.05 were considered statistically significant.

I C van Eijk, M E Tushuizen, A Sturk, B A C Dijkmans, M Boers, A E Voskuyl, M Diamant, G.J. Wolbink, R Nieuwland and M T NurmohamedCirculating microparticles remain associated with complement activation despite intensive anti-inflammatory therapy in early rheumatoid arthritisAnn Rheum Dis published online 16 Nov 2009;


Typical association question

Research question: is there an association between age and pain in patients with …?

Hypothesis: pain increases in older patients

Y = a + bX + e

Amsterdam Rehabilitation Research Center | Readeage

pain

50

Amsterdam Rehabilitation Research Center | Reade15

Simple (uni) linear regression analysis

Difference with correlation analysis: prediction line that gives the best description of the scatter

plot, best fitting line difficult to draw line by hand solve problem with mathematical equation


Simple (uni) linear regression analysis

We use the ‘Method of Least Squares’ to fit the best line

Minimal distance between the data and the fitting line


pain

50


Simple regression analysis

1 = difference between age 0 and age 1 difference between age 1 and age 2

----------------------------------- difference between age 30 and age 31

Pain = 0 + 1 * age

What is 0?

What is 1? 1 = Beta=b

0 = pain at age is 0

Amsterdam Rehabilitation Research Center | Reade19

Mathematical equation to describe the relationship

y = a + b*x

y is called the dependent (outcome) variable

x is called the independent (predictor, explanatory) variable

a is the intercept: value of y when x=0

b (unstandardized beta) is the ´slope´: it represents the amount by which Y increases on average if we increase x by one unit

a and b are called regression coefficients



Regression coefficient is equal to the difference in the outcome variable when the determinant one unit changes


pain

50

1

1



pain = - 20 + 0,5 * age

What is –20? What is 0,5?


You can also analyse difference between two groups with simple regression analysis.


Back to 2 groups and analysis of pain

group pre after

medication 75.8 (6.8) 65.8 (10.1)

placebo 75.4 (7.1) 68.2 (9.0)


Group Statistics

100 -7,2000 3,75513

100 -10,0000 6,81650

groepplacebo

nieuwe medicatie

VERSCHILN Mean Std. Deviation

Independent Samples Test

3,598 198 ,000 2,8000 1,26530 4,33470VERSCHILt df Sig. (2-tailed)

MeanDifference Lower Upper

95% ConfidenceInterval of the

Difference

t-test for Equality of Means


Now analysed by simple regression analysis

placebo Medication 1

Continuous outcome

Pain

Amsterdam Rehabilitation Research Center | Readeplacebo Medication 1

Continuous outcome

Pain



Regression coefficient is equal to the difference in mean between two comparable groups


Simpe (uni) linear regression analysis

1 = mean difference between placebo and medication

Pain = 0 + 1 * group

placebo = 0; medication = 1

0 = mean in controlegroup


Coefficientsa

-7,200 ,550 -13,084 ,000

-2,800 ,778 -3,598 ,000

(Constant)

groep

Model1

B Std. Error


t Sig.

Dependent Variable: VERSCHILa.

0 1


Hypothesis test for β

SEt

N-2 degrees of freedom


P value?

t -3,598

778,0

800,2t


Back to the exampleExperimental designIncluding another medicineThree comparable groups


Pain

group To T1

medication1 75.8 (6.8) 65.8 (10.1)

medication2 76.8 (7.5) 61.9 (11.7)

placebo 75.4 (7.1) 68.2 (9.0)

Amsterdam Rehabilitation Research Center | Readeplacebo medication1 medication2

Continuous outcome


Coefficientsa

-6,850 ,586 -11,681 ,000

-3,850 ,454 -8,475 ,000

(Constant)

groep

Model1

B Std. Error


t Sig.


Group analysed as continuous variabele


But…, group isn’t a continous variable: a categorical variable Therefore it needs to be analysed by dummy-variables



Dummy variables

Categorical Variables Codings

,000 ,000

1,000 ,000

,000 1,000

placebo

nieuwe medicatie

alternatieve medicatie

GROEP(1) (2)

Parameter coding

Dummy 1: new medication - placebo

Dummy 2: alt. medication - placebo

Placebo: controle / control groep



Pain = 0 + 1 * medicationgroup1 + 2 * medicatiogroup2

What is 0?0 = mean of placebogroupWhat is 1?1 = difference between placebo and medication1What is 2?2 = difference between placebo and medication2

Amsterdam Rehabilitation Research Center | Readeplacebo medication1 medication2

Continuous outcome


Coefficientsa

-7,200 ,642 -11,222 ,000

-2,800 ,907 -3,086 ,002

-7,700 ,907 -8,487 ,000

(Constant)

DUMMIE1

DUMMIE2

Model1

B Std. Error


t Sig.


Pain = 0 + 1 * medicationgroup1 +

2 * medicationgroup2


Intermezzo

Little excercise..•

Is there a relationship between your height (cm) and shoesize (european size)…

•Estimate relationcoefficient…

•What does that mean?

•Estimate formula Height = ? + ? * shoesize

•Group: men/woman.

•Groups: occupational therapy, physiotherapy, other.


Assumption linear regression analysis

Linear relationship between x en y•S

catter diagram(

otherwise Logaritmic transformation (next week)

For each value of x, there is a distribution of values of y in the population; this distribution is Normal

•Analyses of the residuals

Variability of the distribution of y values in the population is the same for all values of x, i.e. the variance is constant (s2 / sd)

•Analyses of the residuals


Checking for linearity

Scatterplot

Adding a quadratic term

Splitting exposure variable into groups (4-5)


Adding a quadratic termpain

age


Checking for linearity

Splitting exposure variable into groups


Splitting exposure variable into groupspain

age

1

2

34


Example in SPSS

Examine the association between age and pain score at baseline.

ScatterplotLinear regression analysisChecking for linearity

•Adding a quadratic term

•Splitting exposure variable into groups


Scatter plot

40,00 50,00 60,00 70,00 80,00 90,00

age

65,00

70,00

75,00

80,00

pa

in


Lineair regression analysis

Pain (at baseline) = 56.2 + 0.23 * age

Coefficientsa

56,239 2,131 26,394 ,000

,234 ,033 ,523 7,005 ,000

(Constant)

age

Model1

B Std. Error


Beta


t Sig.

Dependent Variable: paina.


Adding a quadratic term

Coefficientsa

56,239 2,131 26,394 ,000

,234 ,033 ,523 7,005 ,000

49,128 13,116 3,746 ,000

,456 ,405 1,020 1,124 ,263

-,002 ,003 -,499 -,549 ,584

(Constant)

age

(Constant)

age

age2

Model1

2

B Std. Error


Beta


t Sig.




Produce categorical age variableRecode to dummy variablesPerform linear regression analysis with dummiesAre the B’s increasing in a linear order with comparable

distance between the dummies?



Coefficientsa

68,382 ,535 127,936 ,000

1,924 ,774 ,223 2,486 ,014

3,346 ,750 ,404 4,459 ,000

5,446 ,768 ,638 7,094 ,000

(Constant)

dummy1

dummy2

dummy3

Model1

B Std. Error


Beta


t Sig.



Questions?

55

Documents

Amsterdam Rehabilitation Research Center | Reade Correlation and linear regression analysis Martin van der Esch, Phd