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 Proportions

 Proportions

 Single proportion

 Paired proportions

 Two independent proportions

Tests on proportions. 1

Single proportion. 1

Paired proportions. 3

Two independent proportions. 5

Tests onproportions

·Singleproportion

·Pairedproportions

·Two independent proportions

Menu location: Analysis_Proportions.

This section constructsconfidence limits and probabilities for binomialproportions and their differences. Exact tests are employed whereverpossible.

For tests on ratios ofproportions please see exacttests on counts, chi-square testsand miscellaneous.

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Singleproportion

Menu location: Analysis_Proportions_Single.

This function compares anobserved single binomialproportion with an expected proportion.

expected proportion (binomial parameter) = pi

successes = r

observations/trials = n

observed proportion p = r / n

The expected proportion (pi) isthe probability of success on each trial, for example pi = 0.5 for coming upheads on the toss of a coin. The sign testis basically a single proportion test based on pi = 0.5. Some authors refer tothis method as a "binomial test".

StatsDirect provides an exact confidence interval and an approximate mid-Pconfidence interval for the single proportion. You are also given exact P andexact mid-P hypothesis tests for the proportion in comparison with an expectedproportion, i.e. null hypothesis that p = pi (Armitage and Berry,1994; Gardner and Altman, 1989).

Assumptions:

·two mutuallyexclusive outcomes

·random sample

Consider using mid-P values and intervalswhen you have several similar studies to consider within an overallinvestigation (Armitageand Berry, 1994; Barnard, 1989).

Technical Validation

The Clopper-Pearsonmethod is used for the exact confidence interval and the Newcombe-Wilsonmethod is used for the mid-P confidence interval (Newcombe, 1998c).Mid-P probabilities are found by subtracting the exact probability for theobserved count from the cumulative total; this subtraction is done on each sidefor the two sided result.

If n is greater than one millionthen the normal approximation to the binomial distribution is used to calculatethe P values, otherwise exact cumulative binomial probabilities are given.

Example

From Armitage and Berry(1994, p. 119).

In a trial of two analgesics, Xand Y, 100 patients tried each drug for a week. The trial order was randomized.65 out of 100 preferred drug Y.

To analysethese data in StatsDirect you must select singleproportion from the proportions section of the analysis menu. To select a 95%confidence interval just press enter when you arepresented with the confidence interval menu. Enter n as 100 and r as 65. Enterthe binomial test proportion as 0.5, this is because you would expect 50% of aninfinite number of patients to prefer drug Y if there was no difference betweenX and Y.

For this example:

Total = 100, response = 65

Proportion = 0.65

Exact (Clopper-Pearson)95% confidence interval = 0.548151 to 0.742706

Using null hypothesis that thepopulation proportion equals .5

Binomial one sided P = .0018

Binomial two sided P = .0035

Approximate (Wilson) 95% mid P confidence interval =0.552544 to 0.736358

Binomial one sided mid P = .0013

Binomial two sided mid P = .0027

Here we can conclude that thepropwww.med126.comortion was statistically significantly different from 0.5. With 95%confidence we can state that the true population value for the proportion liessomewhere between 0.55 and 0.74.

P values

confidenceintervals

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Pairedproportions

Menu location: Analysis_Proporti医.学全在线ons_Paired.

This function examines thedifference between a pair of binomialproportions.

Two proportions are paired (asopposed to independent) if they share a common feature that affects theoutcome. For example, when comparing two laboratory methods (culture media) todetect bacteria in samples of blood, if blood from the same sample is put intoboth methods, this is the "pairing". Pairs of results from multiplesamples can then be compared as a pair of proportions:

total n = a+b+c+d

responding in both categories = r = a

responding in first category only = s = b

responding in second category only = t = c

proportion 1 = (r + s) / n

proportion 2 = (r + t) / n

proportion difference (delta) = (s - t) / n

Another way of looking at thesedata is to examine how the grouping into category A depends upon the groupinginto category B (see exact test formatched pairs of counts).

StatsDirect gives you exact and exact mid-P hypothesis tests for the equalityof the two proportions (i.e. delta = 0) and gives you a confidence interval forthe difference between them.

Assumptions:

·two mutuallyexclusive outcomes

·random sample fromone population

Consider using mid-P values and intervals when you have severalsimilar studies to consider within an overall investigation (Armitage and Berry,1994).

Technical Validation

Exact methods are used throughout(Armitage andBerry, 1994; Liddell, 1983). The two sided exact P value equates with theexact test for a paired fourfoldtable (Liddell,1983). With large numbers an appropriate normal approximation is used inthe hypothesis test (note that most asymptotic methods tend to mid-P).

The confidence interval isconstructed using Newcombe's refinement of Wilson's score based method, this is close to a mid-P interval (Newcombe, 1998a).

Example

From Armitage and Berry(1994, p. 138).

The data below represent acomparison of two media for culturing Mycobacterium tuberculosis. Fifty suspectsputum specimens were plated up on both media and the following results wereobtained:

To analysethese data in StatsDirect you must select pairedproportions from the proportions section of the analysis menu. Select a 95%confidence interval by pressing enter when you are presented with theconfidence interval menu. Enter TOTAL (n) as 50, BOTH (k) as 20, FIRST (r) as12 and SECOND (s) as 2.

For this example:

Total = 50, both = 20, first only= 12, second only = 2

Proportion 1 = 0.64

Proportion 2 = 0.44

Proportion difference = 0.2

Exact two sided P = .0129

Exact one sided P = .0065

Exact two sided mid P = .0074

Exact one sided mid P = .0037

Score based (Newcombe)95% confidence interval for the proportion difference:

0.056156 to 0.329207

Here we can conclude that theproportion difference is statistically significantly different from zero. With95% confidence we can say that the true population value for the proportiondifference lies somewhere between 0.06 and 0.33. This leaves us with littledoubt that medium A is more effective than medium B for the culture of tuberclebacilli.

Compare these results with theexact test for matched pairs.

P values

confidenceintervals

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Twoindependent proportions

Menu location: Analysis_Proportions_Two Independent.

This function examines thedifference between two independent binomialproportions.

Another way of looking at twoproportions is to put the counts/frequencies into a 2 by 2 contingency tableand examine the relationship between the grouping into rows and the groupinginto columns (see Fisher'sexact test and 2 by 2chi-square test).

r1 = a

r2 = b

n1 = a+c

n2 = b+d

p1 = r1/n1

p2 = r2/n2

proportion difference (delta) = p1-p2

For example, the proportion ofstudents passing a test when taught using method A could be compared with theproportion passing when taught using method B.

StatsDirect provides an hypothesis test for theequality of the two proportions (i.e. delta = 0) and a confidence interval forthe difference between the proportions. An exact two sided P value iscalculated for the hypothesis test (null hypothesis that there is no differencebetween the two proportions) using a mid-P approach to Fisher'sexact test. The conventional normal approximation is also given for thehypothesis test, you should only use this if the numbers are large and theexact (mid) P is not shown (Armitage and Berry,1994).

Assumptions:

·two mutuallyexclusive outcomes

·two random samples

·samples from two independent populations

Technical Validation

The iterative method of Miettinen and Nurminen is used toconstruct the confidence interval for the difference between the proportions (Mee, 1984; Anbar,1983; Gart and Nam, 1990; Miettinen and Nurminen, 1985; Newcombe, 1998b).This "near exact" confidence interval will be in close but not inexact agreement with the exact two sided (mid) P value; i.e. just excludingzero just above P = 0.05.

Example

From Armitage and Berry(1994).

Two methods of treatment, A andB, for a particular disease were investigated. Out of 257 patients treated withmethod A 41 died and out of 244 patients treated with method B 64 died. We wantto compare these fatality rates.

To analysethese data in StatsDirect you must select unpairedproportions from the proportions section of the analysis menu. Enter totalobservations in sample 1 as 257, number responding in sample 1 as 41, totalobservations in sample 2 as 244 and number responding in sample 2 as 64.

For this example:

Total 1 = 257, response 1 = 41

Proportion 1 = 0.159533

Total 2 = 244, response 2 = 64

Proportion 2 = 0.262295

Proportion difference = -0.102762

Approximate (Miettinen)95% confidence interval = -0.17432 to -0.031588

Exact two sided (mid) P = .0044

Standard error of proportiondifference = 0.03638

Standard normal deviate (z) =-2.824689

Approximate two sided P = .0047

Approximate one sided P = .0024

Here we can conclude that thedifference between these two proportions is statistically significantlydifferent from zero. With 95% confidence we can state that the true populationfatality rate with treatment B is between 0.03 and 0.17 greater than withtreatment A.

P values

confidenceintervals