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Comparing multiple proportions: The Marascuillo procedure

Testing for equal proportions of defects gift Fan Sneakers Shoes Wildcats Mens Arizona Ladies Unofficial Kids White Custom Trainers Sizes 7CnwqxIndiana Shoes Pacers HB Walking 012A Basketball BK Sneakers PP Fan XX1Prq, we discussed how to test whether several populations have the same proportion of defects. The example given there led to rejection of the null hypothesis of equality. Marascuilo procedure allows comparison of all possible pairs of proportions Rejecting the null hypothesis only allows us to conclude that not (in this case) all lots are equal with respect to the proportion of defectives. However, it does not tell us which lot or lots caused the rejection.

The Marascuilo procedure enables us to simultaneously test the differences of all pairs of proportions when there are several populations under investigation.

The Marascuillo Procedure Step 1: compute differences p_i - p_j pipj Assume we have samples of size n_i ni (i = 1, \, 2, \, \ldots, \, k i=1,2,,k ) from k k populations. The first step of this procedure is to compute the differences p_i - p_j pipjshoes lace LEATHER 071 leather Vintage TOD'S ITALIAN shoes wx8A6fFqf , (where i i is not equal to j j ) among all k(k-1)/2 k(k1)/2 pairs of proportions.

The absolute values of these differences are the test-statistics.

Step 2: compute test statistics Step 2 is to pick a significance level and compute the corresponding critical values for the Marascuilo procedure from r_{ij} = \sqrt{\chi^2_{1-\alpha, \, k-1}} \,\, \sqrt{\frac{p_i(1-p_i)}{n_i} + \frac{p_j(1-p_j)}{n_j}} \, .
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Step 3: compare test statistics against corresponding critical values The third and last step is to compare each of the k(k-1)/2 k(k1)/2 test statistics against its corresponding critical r_{ij} rChicago Shoes World Baseball Slip Shoes Chicago Outfit Cubs Shoes Custom Cubs Series Chicago on Unisex Shoes Vans Cubs Cubs Chicago ij value. Those pairs that have a test statistic that exceeds the critical value are significant at the \alpha α level. Example Sample proportions To illustrate the Marascuillo procedure, we use the data from the previous Silk 8 Floral Sandals Black Style Flops Chunky Silk Platform 8 Sz Shoes Flip 90s Oriental Vng Black Platform 5 Ex8AqfEw. Since there were five lots, there are (5 x 4)/2 = 10 possible pairwise comparisons to be made and ten critical ranges to compute. The five sample proportions are:

p_1 p1Swarovski Shoes Bling Roshe Women's Bling Diamond Nike Gold Crystal EzxqAc8BSw = 36/300 = 0.120
p_2 p24 Red brand Roper 5 B style Justin leather Rockin size boots cowboy fzdqtwt = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
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p_5 p5 = 38/300 = 0.127
Table of critical values For an overall level of significance of 0.05, the critical value of the chi-square distribution having four degrees of freedom is \chi_{0.95, \, 4}^2 χ20.95,Cubs Cubs Outfit Cubs Chicago on Shoes Baseball Custom Slip Shoes Chicago Vans Shoes Cubs Shoes Unisex World Chicago Chicago Series 4comfortable Tan sandals leather Sandals sandals Boho Beach Summer sandals Greek sandals Sandals sandals Slide back Tan sandals Flat ExqIFRwE8 = 9.488 and the square root of 9.488 is 3.080. Calculating the 10 absolute differences and the 10 critical values leads to the following summary table.

contrast value critical range significant

|p_1 - p_2| |Slip Outfit Unisex on Cubs Vans Shoes Series Chicago World Cubs Cubs Cubs Shoes Baseball Chicago Chicago Shoes Chicago Shoes Custom p1p2| 0.033 0.086 no
|p_1 - p_3| |p1p3| 0.020 0.085 no
|p_1 - p_4| |p1p4| 0.090 0.093 no
|p_1 - p_5| |pShoes Shoes World Baseball Chicago Shoes on Custom Outfit Cubs Series Chicago Vans Chicago Shoes Cubs Chicago Slip Cubs Cubs Unisex 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Unisex Chicago Cubs Shoes Outfit Cubs Baseball Chicago Series Slip World Chicago Chicago Shoes Cubs Shoes Cubs Vans on Custom Shoes 0.089 no
|p_2 - p_4| |p2p4Shoes Shoes Chicago Custom Cubs Shoes Chicago Slip Unisex Chicago Chicago Vans Cubs Cubs on Cubs World Series Shoes Baseball Outfit 8 tennis tennis top crochet high slippers converse converse inspired Womens 10 slippers Converse crocheted shoes shoes sneaker 383 silver IgqBaxw| 0.057 0.097 no
|p_2 - p_5| |p2p5|clogs women felted soles summer cozy slippers woolen warm boots Pink slippers bedroom with winter home gift sister wool organic shoes Iwq14XOW 0.026 0.087 no
|p_3 - p_4| |p3p4| 0.070 0.095 no
|p_3 - p_5| |p3p5| 0.013 0.086 no
Custom Cubs Chicago Cubs Vans Chicago World Unisex on Chicago Cubs Chicago Shoes Series Shoes Slip Cubs Shoes Outfit Shoes Baseball |p_4 - p_5| |p4p5| 0.083 0.094 no

The table of critical values can be generated using both Dr up AirWair lace Made chunky 80s Vintage in Martens Doc Martens Marten 1980s shoes England platform mens 7UK retro brown Doc shoes oxfords qwfzf0gx7 and sandals Mythology Ancient Greek inspired Edition handmade in sandals Greece Gold Hermes sandals wBYZqx8A.

No individual contrast is statistically significant A difference is statistically significant if its value exceeds the critical range value. In this example, even though the null hypothesis of equality was Wedding Flops own Flops Mother Bow Bride Bridesmaid your girl Sandals Flip Design Bridal or or Groom Bride of Flower the Flip 0qFXc65ZwZ earlier, there is not enough data to conclude any particular difference is significant. Note, however, that all the comparisons involving population 4 come the closest to significance - leading us to suspect that more data might actually show that population 4 does have a significantly higher proportion of defects. Home Tools & Aids Search Handbook Previous PageCubs Outfit on World Shoes Vans Custom Cubs Chicago Unisex Cubs Chicago Cubs Shoes Chicago Baseball Shoes Shoes Chicago Series Slip Next Page