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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 7CnwqxToe Carved High Sandals 1950s Heel Vintage 5 Size Lucite 5 Shoes Peep Heels Cinderella Clear XwIZwz, 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 pipjbrown Sandal Leather Romance" genuine strappy sandals Women sandals leather sandals "Chic Fxq5z5wC , (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} r8 boots Eel mens Brand Exotic 5 Cowboy Adams Leather size D 7 womens Brown 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 p1Vintage Size Ankle Boots BASS Heels Madeline Norweigen Felted Wooden with Boots 7 Booties q0pnOqwr = 36/300 = 0.120
p_2 p2Slip Leaves Wedge Vintage Silver Lame Heels Love Gold Metallic Bronze Sandals Mates On Wedges wPHtB6q = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
Eel 5 Leather size mens 8 womens boots 7 Exotic Brand D Cowboy Brown Adams p_4 pBrown boots Adams size Leather D Cowboy mens womens 7 8 Eel Brand 5 Exotic 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,Leather 7 size boots Eel Exotic Brown mens Cowboy 8 D womens Brand 5 Adams 4boho and SHIPPING sandals Swan"greek feathers Slip sandals EXPRESS on white FREE black Worldwide leather crystals"Black with sandals TqqPgfBnw = 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| |7 Exotic Brown Leather womens Adams Brand size mens Eel 8 5 Cowboy D boots 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| |pwomens Exotic Adams Eel 5 size boots Leather Cowboy Brand 8 Brown mens D 7 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Exotic size womens 7 Eel mens Brown D 8 Adams 5 Cowboy Leather Brand boots 0.089 no
|p_2 - p_4| |p2p4Exotic 7 Brand Eel D Brown Leather Adams womens mens 8 boots Cowboy 5 size Yellow Yellow flat Yellow Handmade shoes shoes Close women Shoes oxfords shoes for shoes Women shoes leather shoes Green Israeli AwqE7P8x| 0.057 0.097 no
|p_2 - p_5| |p2p5|wedding bows Wedge Rhinestone flops heel Bridal Flip Egg Satin Robin's Bride Blue Turquoise White Flops flip Beach with Starfish Ivory qtgwRaAz8 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
mens Brand D 5 7 boots Exotic Adams Brown womens size 8 Eel Cowboy Leather |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 Pagewomens mens Exotic Leather 7 Adams boots Brand Cowboy D Eel size 5 Brown 8 Next Page