Topik 6 Kolerasi

8/12/2019 Topik 6 Kolerasi

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CORELATION

- Pearson-r - Spearman-rho


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Scatter Diagram

A scatter diagram is a graph that shows thatthe relationship between two variablesmeasured on the same individual.

Each individual in the set is represented bya point on in the scatter diagram. Thepredictor variable is plotted on thehorizontal axis and the response variable is

plotted on the vertical axis.Do not connect points when drawing ascatter diagram.


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Scatterplot

A scatterplot is a graph that shows locationof each data formed by a pair of X-Y scores.In a positive linear relationship , as the X

scores increase, the Y scores tends toincrease.In a negative linear relationship , as the Xscores increase, the Y scores tends todecrease.In a nonlinear relationship , as the X scoresincrease, the Y scores do not only increaseor only decreases


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Types of relationship

A horizontal scatterplot, with horizontalregression line, indicates no relationship .Slopping scatterplots with regression linesoriented so that Y increases as X increasesindicate a positive linear relationship .Slopping scatterplots with regression linesoriented so that Y decreases as X increases

indicate a negative linear relationship.Scatterplots producing curved regressionlines indicate nonlinear relationships.


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Strength of relationship

The strength of a relationship is the extent to whichone value of Y is consistently paired with one and onlyone value of X.The strength of a relationship is also referred to as thedegree of association between the two variablesThe absolute value of the correlation coefficient (thesize of the number we calculate) indicates the strengthof the relationship.The largest value you can obtain is 1.0 and thesmallest value is 0.The larger the value the stronger the relationship.


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For example, on average, as height in peopleincreases, so does weight.

Height(in) Weight(lbs)

1 60 102

2 62 120

3 63 1304 65 150

5 65 120

6 68 145

7 69 1758 70 170

9 72 185

10 74 210


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Example of a Positive Correlation

If the correlation is positive, when one variable increases, so does the other.


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For example, as study time increases, thenumber of errors on an exam decreases

Studytime (min)

No. Errorson test

1 90 25

2 100 28

3 130 204 150 20

5 180 15

6 200 12

7 220 138 300 10

9 350 8

10 400 6


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Example of a negative correlation

If the correlation is negative, when one variable increases, the other decreases.


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Example of a zero correlation

If there is no relationship between the two variables, thenas one variable increases, the other variable neitherincreases nor decreases. In this case, the correlation iszero. For example, if we measure the SAT-V scores ofcollege freshmen and also measure the circumference oftheir right big toes, there will be a zero correlation.


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What is the correlationcoefficient?

Linear means straight line.Correlation means co-relation, or thedegree that two variables "go together".Linear correlation means to go together in astraight line.The correlation coefficient is a number

that summarizes the direction and degree(closeness) of linear relations between twovariables.


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The correlation coefficient is alsoknown as the Pearson Product-

Moment Correlation Coefficient .The sample value is called r ,

and the population value is called

(rho).


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The correlation coefficient can takevalues between -1 through 0 to +1.

The sign (+ or -) of the correlationaffects its interpretation.When the correlation is positive ( r >

0), as the value of one variableincreases, so does the other.


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Correlation & Association

Scale Example

Interval-interval Pearson r

Ordinal-ordinal Spearman Rank

Nominal-nominal Phi, Chi-square Independent test

Nominal-interval Eta

Nominal-ordinal Theta, Kruskal-Wallis H test

Ordinal-interval Jaspens M, F test


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Pearson correlation coefficient

o The conceptual (definitional) formula ofthe correlation coefficient is:

where x and y are deviation scores, that

SX and SY are sample standard deviations, that is,

(1.1)


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Another way of defining correlation is:

where zx is X in z-score form, zy is Y in z-scoreform, and S and N have their customary meaning.This says that r is the average cross-product of z-scores.

(1.2)


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Where


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Sometimes you will see these formulaswritten as:

and


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These formulas are correct when thestandard deviations used in thecalculations are the estimated populationstandard deviations rather than thesample standard deviations.so the main point is to be consistent.

Either use N throughout or use N-1throughout.


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Example:


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Interpretation of PearsonCoefficient

r Interpretation

0.00-0.20 can be ignored0.20-0.40 low0.40-0.60 medium

0.60-0.80 high0.80-1.00 very high


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Strength of Pearson r

Coefficient Strength

0.01 0.09 Trivial

0.10 0.29 Low to moderate0.30 0.49 Moderate to

substantial

0.50

0.69 Substantial to verystrong0.70 0.89 Very strong

>0.90 Near perfect


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Spearmans Coefficient of RankCorrelation, r s


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Spearmans rank -order correlationcoefficient

The correlation coefficient is used when one or morevariables is measured on an ordinal (ranking) scaleDescribes the linear relationship between two variablesmeasured using ranked scores

Symbol used rs (The subscript s stands for Spearman;Charles Spearman invented this one)


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Computational Formula for the SpearmanRank-Order Correlation Coefficient is:

Rs = 1 6( D2)

-----------N (N2 -1)

N is the number of pair ranks

D is the difference between the two ranks in eachpair


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Running the Spearman Rank-OrderCorrelation Test1. Determine the difference between the ranks for each

subjects2. Square each difference and sum them3. Calculate the rho statistics.4. Compare the obtained rho value with the critical value


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Summary of the Spearman Rank-OrderCorrelation Test

Hypotheses:H0 : Rho = 0Ha : Rho 0, or Rho < 0, or Rho > 0Assumptiojns:Subjects are randomly selected

Observations are ranked orderDecision Rules:n = number of pairs of ranksIf rhoobt rhocrit, reject H0If rhoobt < rho crit, do not reject H0

Formula rho = 1 6( D2)

n (n2 -1)


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Sample data

Participant Observer A: X Observer B: Y

1 4 3

2 1 2

3 9 8

4 8 6

5 3 5

6 5 4

7 6 78 2 1

9 7 9


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SolutionParticipant Observer A: X Observer B: Y D D 2

1 4 3 1 1

2 1 2 -1 1

3 9 8 1 1

4 8 6 2 4

5 3 5 -2 4

6 5 4 1 1

7 6 7 -1 1

8 2 1 1 1

9 7 9 -2 4

D2=18


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SolutionRs = 1 6( D2)

-----------N (N2 -1)

= 1 (6(18))----------9 (92 -1)

= 1 - ((108)/720)

= 1 0.15= + .85


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What does the value of r s tell you?Spearmans rank correlation coefficient is actually derived fromthe product-moment correlation coefficient , such that:

-1 rs 1rs = 0.85 Means that a child receiving a particular ranking fromone observer tended to receive very close to the same rankingfrom other observerrs = +1 means the ranking is in complete agreementrs = 0 means that there is no correlation between the rankingsrs = -1 means that the ranking are in complete disagreement. Infact they are in exact reverse order.


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Exercise:

The marks of eight candidates in English and Mathematics are:

Candidate 1 2 3 4 5 6 7 8

English (x) 50 58 35 86 76 43 40 60

Maths (y) 65 72 54 82 32 74 40 53

Rank the results and hence find Spearmans rankcorrelation coefficient between the two sets of marks.Comment on the value obtained,


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Solution

English(x)

50 58 35 86 76 43 40 60

Maths

(y)

65 72 54 82 32 74 40 53

Rank x 4 5 1 8 7 3 2 6

Rank y 5 6 4 8 1 7 2 3

D -1 -1 -3 0 6 -4 0 3

D2 1 1 9 0 36 16 0 9 D 2=72


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-----------N (N2 -1)

= 1 (6(72))----------8 (82 -1)

= 1 - ((432)/504)

= 1 0.857= .142

Spearmans coefficient of rankcorrelation is 0.142This appears to show a veryweak positive correlation between the English andMathematics ranking


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Tied Ranks

A tied rank occurs when two participants receive thesame rank on the same variable (e.g two person are tiedfor first on variable x)Tied ranks result in an incorrect value of rsResolve (correct) any tied ranks before computing rsTherefore, for each participant at a tied rank, assign the meanof the ranks that would have been used had there not been a tie


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Example

Runner Race X Race Y To resolve ties New Y

A 4 1 Tie uses ranks 1

and 2, becomes 1.5

1.5

B 3 1 Tie uses ranks 1and 2, becomes 1.5

1.5

C 2 2 Becomes 3rd 3

D 1 3 Becomes 4th 4


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Example

Runner Race X New Y D D 2

A 4 1.5 2.5 6.25

B 3 1.5 1.5 2.25

C 2 3 -1 1

D 1 4 -3 9

D2= 18.5


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-----------N (N2 -1)

= 1 (6(18.5))----------4 (42 -1)

= 1 - ((111)/60)

= 1 1.85= - .85

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Topik 6 Kolerasi