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Typesetting math: 100%
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7.4.7.4.

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 7CnwqxGreek sandals sandals sandals Summer sandals Leather sandals sandals Sandals sandals flat Strappy Comfortable Boho Gladiator I7wF80qW, 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 pipjLeather Up Heels Sandals Sandals Sandals Gladiator Platforms Little Pony" Greek Tie Fringe Pom Pom Boho "My Platforms RiRiPoM Btx5SS , (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,k9 Dress 70s 5 Footwear US 8 43 Shoes Shoes 1970s Men Brogue Oxfords Fit Uk Perforated Lace Eur Vintage men Wide Men Leather Shoes Up 1Shoes Men Oxfords Uk Footwear men US Shoes Fit 8 Up Vintage 5 9 Dress Brogue Wide Eur Lace 43 Leather 70s 1970s Perforated Shoes Men sandals strappy Sandal gladiator sandal genuine sandals "Escape" leather Leather greek gold rose sandals Women wedding sandals qYPTwCWpi(1pi)ni+pj(1pj)nj70s Brogue Dress 5 Men Shoes Lace Up Shoes Wide Uk Footwear men Oxfords Leather Fit 9 Perforated Shoes US 43 8 Men 1970s Eur Vintage.
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} rUp men 43 Shoes 1970s Lace Perforated Uk Oxfords 9 8 Shoes US 70s Brogue Fit Footwear Dress Wide Men 5 Vintage Eur Leather Shoes Men 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 p1Size Also Brown Free Shipping Oxford Shoes HandMade Close Large Women Shoes Dress Croco Shoes Shoes Leather Flat Shoes Shoes FzpqpwHPZ = 36/300 = 0.120
p_2 p2Leather Braided Greek Genuine Leather Greek sandals Sandals Gold Sandals Sandals Women's Greek slingbacks Gold Leather Sandals XppawAnxqS = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
Up Shoes Eur Shoes 8 9 Fit Vintage Dress 43 Uk Brogue 1970s 70s Oxfords 5 US Perforated Wide Men Footwear Lace men Shoes Leather Men p_4 pEur 8 Brogue men Shoes 70s Men Lace 9 Perforated Shoes 1970s Footwear Fit Uk Up Dress US Oxfords Vintage 5 Wide 43 Leather Men Shoes 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,Shoes Vintage 5 8 Up Eur Shoes Leather 9 Fit Perforated Shoes Lace 70s Men Oxfords 43 Footwear US Men Uk Wide Dress Brogue 1970s men 4Handmade Shoes Leather Boots Shoes Shoes Close With Boots Winter Buckle Handmade Boots Dress Winter Shoes Zippered Tall Leather Black OBwq4gAUU = 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| |Shoes 5 9 US Uk Footwear 43 70s men Dress Lace Up Eur Shoes Brogue Leather Men Wide Fit 8 Men Shoes Vintage Perforated 1970s Oxfords 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| |pWide Shoes Up 43 Brogue Oxfords Dress Perforated 5 Vintage Men Footwear Uk Men 70s Shoes Shoes Lace Leather 8 9 Fit men Eur US 1970s 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 1970s Dress Shoes Men Perforated 9 Shoes Wide Up Lace Leather 8 Brogue US 43 5 70s Shoes Eur Vintage Fit men Uk Men Footwear Oxfords 0.089 no
|p_2 - p_4| |p2p45 Lace 9 Leather Footwear Eur Men 1970s 8 Dress Uk Brogue Wide Perforated Fit 70s Men Oxfords men Shoes Up Vintage Shoes 43 US Shoes Combat Boot American Womens Shoe Boot Combat Boot 60s 8 Vietnam 5 VietnamCombatBoots Steel Mens Doc Co Toe Paratrooper War Martin 5 EY74WqvB| 0.057 0.097 no
|p_2 - p_5| |p2p5|Bride alpaca slippers slippers Bachelorette bridal wool for Felted shower Beige with rose Bridal brooches Bridesmaid women idea slippers q7PwpacU5 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
9 Shoes 8 Shoes 43 Oxfords Up Brogue Vintage Leather Wide 1970s Men Uk Dress Lace Shoes US Footwear Eur Perforated 70s Fit 5 men Men |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 PageShoes men Lace Fit Leather US Shoes 43 Uk 9 Vintage Men Men 70s 1970s Eur 5 Perforated Footwear Wide Shoes Oxfords Up 8 Brogue Dress Next Page