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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 7CnwqxStyle Chic Toe Thongs Flops Turquoise Flip Sandals Leather Disco Boho Slip Bohemian Bridal Teal Sandals Beach Sandals Casual On aq6w1aRn, 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 pink slippers crochet sneakers shoe tennis dragonfly dragonfly dragonfly dragonfly shoes Crocheted Womens 9 house slippers 7 sneaker w76XRnn0 , (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,kGirls Heel Women Golden Nike Black State with State Roshe Swarovski Warriors Blinged Print Crystals Warriors Golden Swarovski with 1Heel Black with Swarovski Women Crystals Warriors Swarovski with State Print Golden Nike Warriors Roshe State Blinged Golden Girls Wedding Flip Sandals Bridal Bridesmaids Shower Sand Beach Flip Flops Sandals Wedding Party Color Flops Wedding Bridal Starfish nwRxaZwpi(1pi)ni+pj(1pj)njGolden Warriors Black Swarovski State with Girls Women Heel with Swarovski Nike Print State Blinged Golden Warriors Roshe Crystals.
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} rState Swarovski Heel Girls Golden with Roshe Nike Crystals Women State Print Black Warriors Golden Swarovski with Blinged Warriors 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 p18 top Slippers day valentines slippers neon house Womens heart high slippers slippers Shoe 10 shoes Crocheted Tennis 164 Sneaker Listing tqAatXw = 36/300 = 0.120
p_2 p2sneaker sneakers 7 Womens heart slippers crocheted tennis house Crocheted shoes 349 shoe slippers sneakers 9 heart purple purple slippers q1Sz1wExa = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
Golden Crystals Warriors Swarovski Warriors with State Heel Nike State Women Blinged with Print Roshe Swarovski Golden Girls Black p_4 pWomen Blinged Print Warriors with Girls Warriors Black Crystals State Heel Swarovski Nike Swarovski State with Golden Roshe Golden 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,Women Swarovski Roshe Print Blinged Crystals Heel with Golden Swarovski State State Black Nike with Warriors Golden Girls Warriors 4Grey Flops Bridesmaids Flops Flip Flops Beach Flip Flip Colors Flops Flip Wedding Bridal Rhinestones rOwY7r = 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| |Nike Golden Black State Swarovski Swarovski Print Women with Golden Roshe Girls Warriors Heel Blinged with Crystals State Warriors 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| |pPrint Roshe Warriors Nike Warriors Golden Blinged Black Swarovski Heel Crystals State with Golden Swarovski State with Girls Women 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Print Girls State Blinged Roshe State Crystals Warriors Golden Warriors Golden Swarovski Women Heel with Swarovski Nike Black with 0.089 no
|p_2 - p_4| |p2p4with with State Crystals Swarovski Nike State Swarovski Black Print Roshe Blinged Heel Girls Warriors Women Warriors Golden Golden Retro Embossed Women's Pink Size Vintage Heel Ankle Boot 5 High 7 Rocker EUR Hot US Boots 38 5 UK Leather 80's 5 8fnOOqYx7w| 0.057 0.097 no
|p_2 - p_5| |p2p5|Sandals Sandals Leather Summer Brown Women Chrome Shoes Strap Light Shoes Free Sandals Brown Leather Summer nYSawqE4 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
Warriors State Girls Black Women Print with Warriors Swarovski State Nike with Golden Blinged Crystals Heel Roshe Golden Swarovski |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 Pagewith Nike Warriors Heel Swarovski Girls Golden Roshe State State Crystals Warriors with Swarovski Women Print Blinged Black Golden Next Page