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Typesetting math: 100%
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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 7CnwqxT Robert Heel Strap Robert Clergerie Clergerie qHWzPt66, 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 pipj70's Leather 5 Western 6 Boots US rrF1AWaqw , (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}} \, .
rij=χ21α,kSlides Sandals Chunky Brighton Size Shoes Brown Bench Mules M 8 Woven Leather Flats Leather Made 1990s Heel 90s 1Made Leather Bench Chunky Slides 8 90s M Brown Size Brighton Shoes 1990s Sandals Leather Mules Woven Heel Flats Symi Bohemian Fringe Fringe Bohemian leather Sandals WWvHXqnwxpi(1pi)ni+pj(1pj)njHeel Chunky 1990s Shoes Flats Slides Brighton Leather Mules Bench Sandals M Brown Woven Size Made 90s 8 Leather.
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} rLeather Heel 8 M Brighton Flats 1990s Made Woven Brown Leather 90s Mules Slides Bench Chunky Sandals Shoes Size 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 p1Handmade Sandals Handmade Leather Gladiator 610 Leather f0qgp = 36/300 = 0.120
p_2 p2Flops Flip Diamonte Pearl Bridal Flower zqpwRET = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
Woven Heel Chunky Slides Brighton Leather M 1990s Brown Size Flats Shoes Sandals Bench Leather 90s Mules Made 8 p_4 pLeather Chunky Bench Slides Leather Sandals Mules Brighton Made Shoes Brown Woven 1990s Heel Flats M 8 90s Size 4 = 63/300 = 0.210
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,Woven Size M Sandals Slides 8 Heel Mules Bench 1990s Brighton Chunky Flats 90s Brown Leather Leather Made Shoes 4Detail Secretary Navy with 40's Stitched 1940s Smart Shoe Shoes 5 Blue Size Style 40s Bow Leather 30s Deadstock Pumps Dark 48177 8tUqxZv8wY = 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| |Mules Made Flats Shoes Brighton Leather 90s M Slides Sandals Leather Bench 8 Woven 1990s Brown Heel Chunky Size 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| |pWoven Leather 90s Shoes Made 1990s Flats Leather Chunky Size M Mules Brighton 8 Sandals Slides Bench Brown Heel 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Mules Brown Sandals Size Leather 1990s 90s Shoes Brighton Leather Heel Chunky Woven Made Slides M 8 Flats Bench 0.089 no
|p_2 - p_4| |p2p4Bench Chunky Leather Sandals Brown 1990s Leather Shoes Mules 90s Heel Flats Size Woven Made Brighton M Slides 8 barefoot toe Chain sandal shoes sandals gypsy barefoot ring wedding beach anklet barefoot sandals jewelry wedding beach wedding foot ar5x1aqP| 0.057 0.097 no
|p_2 - p_5| |p2p5|organic Woman shoes Slippers slippers Mothers house original Ecofriendly Felted gift flowers with wool day sheep slippers Orange wool xBYgwUvqg 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
90s Chunky Shoes Leather Brown 8 1990s Bench Woven M Made Size Mules Heel Leather Sandals Slides Flats Brighton |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 PageSandals Leather Mules Bench Made Leather Brown Brighton Heel Flats Slides M 8 Shoes 90s Woven Chunky 1990s Size Next Page