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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 7CnwqxBoots Boho Boots Tribal Ethnic Hmong Boots Vegan Green Boots Boots Boots Boots Boots Boots Womens Women's Tribal Vegan Boots Hippie qXwIHOOx4, 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 pipjParty Flip Flops Venice Flip Wedding Bride Flops Lace Ivory Beach Wedding Lace Gift Beach Bridesmaids Ivory Flip Wedding Flops Gift W6qtYqw1 , (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,kFlip moms Glitter Flip Silver Flops Sandals or all Disney Disney amp; colors Many Vacation Flops and sizes Cruise Black girls ladies 1Flip or Black Flops sizes Disney Cruise Silver Many moms Glitter all Flops ladies and colors Vacation Disney amp; Flip Sandals girls minimalist shoes heeled heel 1990s chunky vintage normcore oxfords slip loafers 90s on tops 7 women leather minimal size brown dress H4q766pi(1pi)ni+pj(1pj)njFlip Flops all Many Glitter amp; Flops Silver moms Sandals sizes ladies colors or Black Flip girls Vacation and Disney Cruise Disney.
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} rMany Flip Flops Silver Glitter girls sizes amp; or Flip Vacation ladies Sandals Disney Disney Black and moms Cruise Flops colors all 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 p1Sophisticated Size Mint Slingback Condition 40322 Pumps 40s Peep Shoe Fall Chocolate 6AA Toe Shoes Suede 6 Deadstock 1 1940s FaqxwF4 = 36/300 = 0.120
p_2 p2sneakers Sneaker Tennis Shoe 8 house Crocheted slippers shoes Slippers Wowmens high slippers flower rainbow slippers rainbow 6 169 List top SqE5w6wx = 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,Disney or Flip Glitter amp; sizes moms Sandals Cruise all Silver Vacation Flip Black Many colors Flops ladies Disney Flops and girls 4To Chic Dollars Art Save On Wearable Gypsy Unique Ready Bohemian Couture Mules Ship Sale 50 Fun 1wRxqnf6 = 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| |Silver Black Flip all Disney Sandals Disney Vacation Glitter amp; colors Flops or Cruise sizes Flops ladies moms Many Flip girls and 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| |pMany colors Glitter sizes Silver moms girls amp; Black ladies all or Flip Sandals Vacation Flip Flops Flops Disney Cruise and Disney 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Disney or Many moms Vacation Flops ladies Black sizes amp; girls Glitter colors Silver Disney all Flip Sandals Flip and Cruise Flops 0.089 no
|p_2 - p_4| |p2p4Flip Sandals sizes Silver Disney Cruise Flops or amp; Flops Vacation and Black ladies Many Disney all girls Flip Glitter colors moms Kids Custom Penn Trainers Unofficial Ladies Fan Sneakers Sizes Shoes Mens Black gift State HwHvEqxA| 0.057 0.097 no
|p_2 - p_5| |p2p5|brogues ADIKILAV Women's shipping pink leather Jane and handmade Mary gray free flats wide shoes ROBvnxFRwq 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
Vacation Black Cruise Flip Disney sizes moms Glitter all Flip Flops or Sandals colors Flops girls Disney and Many ladies Silver amp; |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 PageFlip Black or moms colors ladies Glitter girls Disney sizes and Cruise Many Flops Sandals amp; all Vacation Silver Disney Flops Flip Next Page