topik 6 kolerasi
Post on 03-Jun-2018
242 Views
Preview:
TRANSCRIPT
-
8/12/2019 Topik 6 Kolerasi
1/42
CORELATION
- Pearson-r - Spearman-rho
-
8/12/2019 Topik 6 Kolerasi
2/42
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.
-
8/12/2019 Topik 6 Kolerasi
3/42
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
-
8/12/2019 Topik 6 Kolerasi
4/42
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.
-
8/12/2019 Topik 6 Kolerasi
5/42
-
8/12/2019 Topik 6 Kolerasi
6/42
-
8/12/2019 Topik 6 Kolerasi
7/42
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.
-
8/12/2019 Topik 6 Kolerasi
8/42
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
-
8/12/2019 Topik 6 Kolerasi
9/42
Example of a Positive Correlation
If the correlation is positive, when one variable increases, so does the other.
-
8/12/2019 Topik 6 Kolerasi
10/42
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
-
8/12/2019 Topik 6 Kolerasi
11/42
Example of a negative correlation
If the correlation is negative, when one variable increases, the other decreases.
-
8/12/2019 Topik 6 Kolerasi
12/42
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.
-
8/12/2019 Topik 6 Kolerasi
13/42
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.
-
8/12/2019 Topik 6 Kolerasi
14/42
What is the correlationcoefficient?
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).
-
8/12/2019 Topik 6 Kolerasi
15/42
What is the correlationcoefficient?
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.
-
8/12/2019 Topik 6 Kolerasi
16/42
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
-
8/12/2019 Topik 6 Kolerasi
17/42
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)
-
8/12/2019 Topik 6 Kolerasi
18/42
Pearson correlation coefficient
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)
-
8/12/2019 Topik 6 Kolerasi
19/42
Pearson correlation coefficient
Where
-
8/12/2019 Topik 6 Kolerasi
20/42
Pearson correlation coefficient
Sometimes you will see these formulaswritten as:
and
-
8/12/2019 Topik 6 Kolerasi
21/42
Pearson correlation coefficient
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.
-
8/12/2019 Topik 6 Kolerasi
22/42
Example:
-
8/12/2019 Topik 6 Kolerasi
23/42
-
8/12/2019 Topik 6 Kolerasi
24/42
-
8/12/2019 Topik 6 Kolerasi
25/42
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
-
8/12/2019 Topik 6 Kolerasi
26/42
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
-
8/12/2019 Topik 6 Kolerasi
27/42
Spearmans Coefficient of RankCorrelation, r s
-
8/12/2019 Topik 6 Kolerasi
28/42
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)
-
8/12/2019 Topik 6 Kolerasi
29/42
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
-
8/12/2019 Topik 6 Kolerasi
30/42
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
-
8/12/2019 Topik 6 Kolerasi
31/42
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)
-
8/12/2019 Topik 6 Kolerasi
32/42
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
-
8/12/2019 Topik 6 Kolerasi
33/42
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
-
8/12/2019 Topik 6 Kolerasi
34/42
SolutionRs = 1 6( D2)
-----------N (N2 -1)
= 1 (6(18))----------9 (92 -1)
= 1 - ((108)/720)
= 1 0.15= + .85
-
8/12/2019 Topik 6 Kolerasi
35/42
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.
-
8/12/2019 Topik 6 Kolerasi
36/42
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,
-
8/12/2019 Topik 6 Kolerasi
37/42
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
-
8/12/2019 Topik 6 Kolerasi
38/42
SolutionRs = 1 6( D2)
-----------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
-
8/12/2019 Topik 6 Kolerasi
39/42
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
-
8/12/2019 Topik 6 Kolerasi
40/42
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
-
8/12/2019 Topik 6 Kolerasi
41/42
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
-
8/12/2019 Topik 6 Kolerasi
42/42
SolutionRs = 1 6( D2)
-----------N (N2 -1)
= 1 (6(18.5))----------4 (42 -1)
= 1 - ((111)/60)
= 1 1.85= - .85
top related