Concepts of Statistics
Concepts of Statistics STAT 135
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This 5 page Class Notes was uploaded by Floy Kub on Thursday October 22, 2015. The Class Notes belongs to STAT 135 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 29 views. For similar materials see /class/226732/stat-135-university-of-california-berkeley in Statistics at University of California - Berkeley.
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Date Created: 10/22/15
TWO WAY ANOVA Next we consider the case when we have two factors categorizations eg lab and manufacturer If there are I levels in the rst factor and J levels in the second factor then we can think of this situation as one where there are I x J levels of the combined factors Notation Notation wise we simply add another subscript to the response that is y now has a triple subscript where gig1 represents the measurement on the kth subject that belongs to both the ith group lab of the rst factor and thej group manufacturer of the second factor i 1 I j 1 J andk1nj For simplicity we will only work with the special case of mm K ie all subgroups have the same number of responses Then we write the model as follows yam 04 m ELM So when i 1 andj 1 gm quot7 E1 and when i 2 and j 1 3411 E2 Again we need a constraint because our model is over parameterized We add the constraint that 2139 2739 77M 0 A Simpler Submodel In our example of the study of the measurement process we nd that with 7 labs and 4 manufac turers we have 28 levels If the effect of the lab is the same regardless of which manufacturer the tablets are coming from and if the effect of the manufacturer is the same regardless of which lab is measuring the tablets then we could express the model as yam 04 Bi W ELM Note that now we have only I J levels rather than I x J This model is called an additive model It puts structure on the levels That is the difference between measurements at LAb 1 and Lab 2 of tablets from Manufacturer A is 32731 and this difference is the same for the measurements at Labs 1 and 2 for tablets from Manufacturer B ie there is no interaction between lab and manufacturer Degrees of Freedom To see that the additive model is a submodel of the full model we can we express the full model as follows yam 04 Bi v 113 ELM Now again we need to put constraints on the parameterization If we think about it from the geometric perspective we see that the 1 vector lies in both the space spanned by the lab indicators the ei and the space spanned by the manufacturer indicators the uj So the 1 vector and If 1 of the e vectors and J7 1 of the uj vectors are all that is needed for the additive part of the model As for the rest suppose we have vectors VM that indicate whether a response belongs in group ij or not Note that 2 VM i and and that Zivm39 So we need only W of these I x J vectors All together that gives us 1 I 7 1 7 W W W of the 1 I J IJ vectors If we are to put all of the parameters in then we must add constraints Traditionally these are Z W 0for Z W 0for How many constraints do we have i Sums 0f Squares The Anova table of the sums of squared deviations helps us assess whether the simple additive model is adequate to describe the variation in the means and whether there is a lab effect or a manufacturer effect ie whether all of the B 0 or all of the W 0 The decomposition of the sums of squares is a bit more complex here First we need to introduce some more notation 1 1 9i 3yijk7fori1l Now let s look at the sums of squares 22y 727 l To begin let s add and subtract the J means 3 z 7 i j k i 92 i 2 Z 223139739 92 39 k i j k Show that the cross product term is 0 We call the rst sum on the right hand side of the equation the error sum of squares or SSE We want to further decompose the second term 2 k ij 2 2 l 7 What do we add and subtract 2711 or 37ij Both The three terms on the right hand side of the equality are called the interaction sum of squares or SSLM the sum of squares due to Lab or SSL and the sum of squares due to Manufacturer or 5M Show that the cross products are all 0 ANOVA Table Arrange the sum of squares into an ANOVA table Source DF Sum of Squares Mean Square F statistic Labs Manufacturer 3 Interaction 8 Error 60 Total 85 The rst F statistic is used to test whether there is a difference between labs7 ie whether there is a lab effect The second F statistic is used to test whether there is a difference between man ufactureres The third is to test the additive model7 ie is there an interaction between lab and manufacturer
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