research - statistical tool quatros

8/6/2019 Research - Statistical Tool Quatros

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SPEARMAN'S RHO

Spearman's rank correlation coefficient or Spearman's rho,named after Charles Spearman and often denoted by the Greekletter (rho) or as r

s, is a non-parametric measure of statistical

dependence between two variables.

It assesses how well the relationship between two variables can bedescribed using a monotonic function.

If there are no repeated data values, a perfect Spearmancorrelation of +1 or 1 occurs when each of the variables is a

perfect monotone function of the other.

Measure the linear relationship between two variables. It differsfrom Pearson's correlation only in that the computations are doneafter the numbers are converted to ranks.


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The sign of the Spearman correlation indicates the direction of

association betweenX (

the independent variable) andY (

thedependent variable).

IfY tends to increase when X increases, the Spearmancorrelation coefficient is positive.

IfY tends to decrease when X increases, the Spearmancorrelation coefficient is negative.

Oftenly described as being "nonparametric."


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PROCEDURE FOR USING SPEARMAN'S RANK CORRELATION

1. State the null hypothesis i.e. "There is no relationship between thetwo sets of data.

2. Rank both sets of data from the highest to the lowest. Make sure tocheck for tied ranks.

3. Subtract the two sets of ranks to get the difference d.

4. Square the values of d.

5. Add the squared values of d to get Sigma d2.

6. Use the formula Rs = 1-(6Sigma d2/n3-n) where n is the number ofranks you have.


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EXAMPLE

Problem:

Six nursing students' have the following rankings

in Clinical performance and patient satisfactionon the care. Is there any association between theirclinical performance and patients satisfaction on

the care?


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Respondent Clinical

performance

Patient

satisfaction on

the care

1 85.36 89.67

2 82.98 86.71

3 87.76 88.18

4 84.50 85.63

5 84.03 84.56

6 86.33 87.40


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Respondent Clinical

performance

Patient satisfaction on

the care

1 3 1

2 6 4

3 1 2

4 4 5

5 5 6

6 2 3


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Respondent Clinical

performance

Patient

satisfaction

on the care

D1 D2

1 3 1 2 4

2 6 4 2 4

3 1 2 -1 1

4 4 5 -1 1

5 5 6 -1 1

6 2 3 -1 1

N= 6 12


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Solution:

= 1- 6(12)/ 6(36-1) = 1-72/210 = 1 - 0.34 = 0.66

1. If the Rs value...... is -1, there is a perfect negative correlation.

...falls between -1 and -0.5, there is a strong negative correlation....falls between -0.5 and 0, there is a weak negative correlation.

... is 0, there is no correlation

...falls between 0 and 0.5, there is a weak positive correlation.

...falls between 0.5 and 1, there is a strong positive correlation

...is 1, there is a perfect positive correlationbetween the 2 sets of data.

2. If the Rs value is 0, state that null hypothesis is accepted. Otherwise,say it is rejected.


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KENDALL'S TAU

Kendall rank correlation coefficient, commonly referred toas Kendall's tau () coefficient, is a statistic used to measurethe association between two measured quantities. It was developedbyMaurice Kendall in 1938.

Atau test is a non-parametric hypothesis test which uses thecoefficient to test for statistical dependence. Specifically, it is ameasure of rank correlation: that is, the similarity of the orderingsof the data when ranked by each of the quantities.


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ADVANTAGES OF KENDALLS TAU :

The distribution of Kendalls tau has better statistical properties.

The interpretation of Kendalls tau in terms of the probabilitiesof observing the agreeable (concordant) and non agreeable(discordant) pairs is very direct.

In most of the situations, the interpretations of Kendalls tauand Spearmans rank correlation coefficient are very similar andthus invariably lead to the same inferences.

In Spearmans rank correlation coefficient, the measure of rankcorrelation is the more widely used rank correlation coefficient.


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Symbolically, Spearmans rank correlation coefficient is denoted byrs . Spearmans rank correlation coefficient is given by the followingformula:

rs = 1- (6di2 )/ (n (n2-1)), here di in Spearmans rank correlation

coefficient represents the difference in the ranks given to the valuesof the variable for each item of the particular data. This formula ofSpearmans rank correlation coefficient is applied in cases whenthere are no tied ranks. However, in the case of fewer numbers oftied ranks, this approximation of Spearmans rank correlation

coefficient provides sufficiently good approximations.


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au-a

Tau-a statistic tests the strength of association of the cross tabulations.

Both variables have to be ordinal. Tau-a will not make any adjustmentfor ties.

Tau-b

Tau-b statistic, unlike tau-a, makes adjustments for ties and is suitablefor square tables. Values of tau-b range from 1 100% negativeassociation, or perfect inversion to +1 100% positive association, orperfect agreement. A value of zero indicates the absence of association.

Tau-c

Tau-c differs from tau-b as in being more suitable for rectangular tablesthan for square tables.


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A hemodialysis patient's perception regarding HHD was significantly andnegatively associated with both social support, Kendall's Tau-C = -0.288,p = 0.038 and communication, Kendall's Tau-C = -0.353, p = 0.001. This

indicates that when a patient scores high on the Patient PerceptionSurvey, meaning he or she has negative perceptions regarding HHD, thepatient as well has low levels of social support and communication. Theassociations between a hemodialysis patient's perception regarding HHDand all other subscales were not significant. Table 4 shows the strength of

the association between patient perceptions and each of the six subscales,such as physical, communication, ability to maintain self-care, socialsupport, psychological status, and nutritional status.

This study utilized the Patient Perception Survey to obtain measuresof hemodialysis patient perceptions regarding HHD. Results from this

sample of 49 patients found that 46.9% of the patients identifiednegative perceptions of HHD and 53.1% of the 49 patients identifiedpositive perceptions of HHD.


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Tau-c

Kendall's Tau-c, also called Kendall-Stuart Tau-c, is a variant ofTau-

b for larger tables. It equals the excess of concordant over discordant pairs,multiplied by a term representing an adjustment for the size of the table.

Tau-c = (C - D)*[2m/(n2(m-1))]

Where:

m = the number of rows or columns, whichever is smallern = the sample size.

Hemodialysis patients' perceptions of home hemodialysis and self-care

Author: Visaya, Marie Angela Date published: April 1, 2010


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Phi Coefficient

The phi coefficient is a measure of the degree of associationbetween two binary variables. It is a nominal association that isbased on the adjusted chi-square.

In calculating the phi, we divide chi-square by the sample size. Thephi-coefcient was designed for the comparison of truly dichotomousdistributions, i.e., distributions that have only two points on theirscale which indicate some unmeasurable attribute.

Attributes such as living or dead, black or white, accept or reject, andsuccess or failure are examples. It is also sometimes known as theYule Phi coefficient is used for a 22 table and a nominal variable.

Testing of significance of Phi is the same as chi-square.


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FORMULA FOR THE PHI COEFFICIENT


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EXAMPLE

43 persons were asked if they believed that there was any truth inhoroscopes or in the existence of UFOs. The results gave:

UFO

s Some truth No truthTOT

ALMight exist 14 10 24

Dont exist 6 13 19TOT

AL 20 23 N=43

Horoscopes


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Applying the above formula, = 0.266

This value of corresponds to a value of2 of 43

(0.266) 2 = 3.04.This may then be tested against the relevant value of2 for 1 degree of

freedom. An alternative signicance test (rarely used) may beperformed by considering the standard error of. Calculation of thisis laborious but if N is not too small, then 1/ N approximates to it.

Interpretation of the Phi coefficient.

I have general rule of thumb for correlation coefficients and youcan use the same rule for the Phi coefficient.

-1.0 to -0.7 strong negative association.-0.7 to -0.3 weak negative association.-0.3 to +0.3 little or no association.+0.3 to +0.7 weak positive association.

+0.7 to +1.0 strong positive association.


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ANOTHER EXAMPLE TRY TO COMPUTE:


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CRAMER'S V

Cramer's Vis the most popular of the chi-square-based measuresof nominal association because it gives good morning from 0 to 1regardless of table size, when row marginals equalcolumn marginals. It was named after the Swedishmathematician and statistician Harald Cramr, who sought tomake statistics mathematically rigorous, much like Kolmogorov'saxiomatization of probability theory.

Significance testing of the Cramers V is the same as the chi-

square test where it says that there is a significant relationshipbetween variables, but it does not say just how significant andimportant this is. Cramer's V is a post-test to give this additionalinformation.


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The value of the Cramers V can be reached at 1 if two variables havean equal marginal. The value of Cramers V is always less than thephi coefficient. The following formula is used to calculate the value

of Cramers V:

Where v is Cramers V and n and m are the sample size and time.

where min is a single value, the smaller of the two quantities (r-1)or (

c-1).


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Cramers V was calculated as a measure ofthe association between IT sophistication for resident caremanagement and the characteristics of ownership, bedsize, and

location.

Interpretation:V may be viewed as the association betweentwo variables as a percentage of their maximum possiblevariation.V2 is the mean square canonical correlation between

the variables. For 2-by-2 tables,V

= phi (hence some packageslike Systat print V only for larger tables).

Meaning of association: V defines a perfect relationship as onewhich is predictive or ordered monotonic, and defines a null

relationship as statistical independence, as discussed in thesection on association. However, the more unequalthe marginals, the more V will be less than 1.0.

Symmetricalness: V is a symmetrical measure. It does not matter

which is the independent (column) variable.


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Data level: V may be used with nominal data or higher.

Other features: V can reach 1.0 only when the two variableshave equal marginals.

Equalmarginals. Cramer's V and all measures which define a

perfect relationship in terms of strict monotonicity requirethat the marginal distribution of the two variables be equalfor the coefficient to reach 1.0.

Following convention, the strength of the relationship will be

interpreted as follows:

values between 0.0-.30 indicate a weak association. values between .31-.60 indicate a moderate association. values >.60 indicate a strong association.


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