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Lecture 16

ADM2304 Lecture 16: 16

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University of Ottawa
Tony Quon

Candidate for one-way ANOVA as we have one factor – gasoline type – and one response variable – gas mileage But a potentially confounding factor that impacts gas mileage is vehicle type If we did a completely randomized design, our samples for each factor may have different types of vehicle which would bring into question any conclusion we might draw One way of controlling for this is to divide the population of cars up by size and randomly assign one car from each size to each gasoline type (essentially creating a 2nd factor) In this way we ensure that the size of the car does not have an impact on the outcome This is called a randomized block design How Does this Extend Matched Pairs? Suppose there were only two gasoline types – regular and premium The matching would be based on vehicle type so that we would make sure that each sample has equivalent vehicle types in it so that we can be sure that the vehicle type is not affecting the results Thus, each SUV in the sample given regular gasoline would be “matched” with an SUV in the sample given premium gasoline Rather than “before” and “after”, we have matching based on characteristics (in this case vehicle type) The randomized block design extends this to more than two populations The model: We proceed essentially as if this were a two-factor analysis with no interaction term Thus, our model is: X =ij +α + β +εi j ij We divide the total variation SSTotal into variation due to the treatment (column), SSB, the variation due to the block (row), SSA and the random variation due to “noise”, SSE SSTotal = SSB + SSA + SSE Degrees of freedom are: N - 1 for SSTotal b - 1 for SSB a - 1 for SSA (a - 1)(b - 1) for SSE where: a is the number of blocks – that is the number of observations in each treatment level b is the number of treatment levels We are only interested in the main effect of the column variable Hypothesis Test: Hypotheses: H0: Main effects due to treatments are all the same (i.e., gasoline has no impact on gas mileage) Ha: Main effect due to some treatment is different from another (i.e., at least one gasoline does improve gas mileage over another) We compare the variation between the treatments means, MSB, with the overall variation, MSE by calculating F = MSB/MSE We reject the null hypothesis if the F-statistic is larger than the critical value Fα with b - 1 and (b - 1)(a - 1) degrees of freedom We could of course also do a test to determine if the mean effect of all the block levels is equal but we would expect to reject that hypothesis as we specifically chose the blocks because we thought they would impact on the results Bonferroni Method for Paired Comparisons We use the Bonferonni margin of error for examining pairs of treatment means: 
 ⎛ 1 1 ⎞ t* MSE ⎜ + ⎟ ⎝ a a ⎠ where a is the number of observations comprising each treatment mean and t* represents the t-value with /2J probability in the right tail Example: Tyco Valves Tyco uses a manufacturing resource planning system to reduce lead-time for manufacturing. It wishes to determine whether lead-times differ according to the type of valve Tyco recognizes that the day of the week the valve is ordered may have an impact on the lead time (especially if it is near the weekend) Thus they look to do a randomized block design to determine the impact of valve type on lead time while controlling for any day of the week effect Valve Type Day Safety Butterfly Clack Slide Poppet NeedleBlock Means Monday 1.6 2.2 1.3 1.8 2.5 0.8 1.7 ANOVA Table Tuesday 1.8 2.0 1.4 1.5 2.4 1.0 1.68 SourceFD SS MS F P-value Wednesday 1.0 1.8 1.0 1.6 2.0 0.8 1.37 Day 4 0.77667 0.19417 4.90.006 Thursday 1.8 2.2 1.4 1.6 1.8 0.6 1.57
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