IX. Complete Block Designs (CBD s)

IX. Complete Block Designs (CBD s) A.Background Noise Factors nuisance factors whose values can be controlled within the context of the experiment but not outside the context of the experiment Covariates - nuisance factors whose values can not be controlled but can be measured within the context of (prior to or during) the experiment) Block - nuisance factors whose values can not be measured or controlled within the context of the experiment (and can not be conveniently measured) Blocks: - are usually groups of experimental subjects who are similar in a manner that may effect the response - are b groups of experimental units that are (relatively) homogenous within and heterogeneous between wrt the blocking factor - are often used to represent a poorly or inexactly measured covariate - are particularly popular in agricultural, scientific, and manufacturing problems (where factors such as plot of land, location, and production lot can not be measured but must be considered) - do not have to be of equal size (but are easier to work with when they have common size k) and will be represented with the subscript h

B. Types of Block Designs - Complete Block Design random allocation of the same number of experimental units per block to each treatment (this requires k = c, c R) so that n hi = k/ = s - Randomized Complete Block Design random allocation of one experimental unit per block to each treatment (this requires k = ) - General Complete Block Design random allocation of the same number of experimental units per block to each treatment (this requires k = c, c R, c > 1) - Confounding result of a design for which all experimental units from each block are assigned to a separate treatment - Incomplete Block Design random allocation of different numbers of experimental units per block to each treatment (this requires k c, c R) - Partial Block Design random allocation where at least one block does not contribute any experimental units to at least one treatment (this requires k < ) the term Incomplete Block design is often reserved for this situation

C. The Randomized Complete Block Design We have treatments (which may be factorial treatment combinations) and b blocks of k = (relatively) homogenous within and heterogeneous between groups (so s = 1 and r i = b) Now let A be a factor with levels i = 1,,a B be a blocking factor with levels h = 1,,b Y hi = the response obtained from the h th block randomly assigned of the i th treatment hi = the error variable (or variation attributable to randomness) The basic model (often called the block-treatment model) is now Y = µ + θ + τ + hi h i hi We again need to make some assumptions about the error terms hi in order to statistically analyze the results of a RCBD experiment: - hi ~ N(0, σ ) - hi s are mutually independent, h = 1,,b, i = 1,, We can test these assumptions using techniques discussed earlier.

To understand the analysis of variance of an RCBD, note that the model looks similar to a two-way main effects model. However, the blocks are not random, but are intentional and deliberate. This generates controversy about testing hypotheses about statistically testing equality of block effects. This leaves one standard hypothesis for an RCBD: H : τ = τ = L = τ 0 1 i.e., there is no treatment effect. Note that this can be partitioned into hypothesis tests about various factors in more complex designs. To test this hypothesis, we need to develop formulas for the various sums of squares. The similarity to the two-way main effects models extends to sums of squares formulae: - the sum of squares for treatments is ( ) y i y sst = b y i - y = - b b i=1 i=1 - the sums of squares for blocks is b b ( ) yh y ssθ = yh - y = - b h=1 h=1

- the total sum of squares is y sstot = y - y = y - b b ( hi ) hi h=1 i=1 i=1 - these sums of squares can be used to find the sum of squares for error: sse = sstot - ssθ - sst Now if H 0 is true, then sst ~ χ σ -1 sse and since ~ χ b--b+1 σ and sst and sse are independent, the ratio of these two Chi-Square statistics (divided by their respective degrees of freedom) yields ssa sst ( -1) σ ( -1) mst = = ~ F sse sse mse ( b -b- +1) σ ( b -b- +1) which provides us with the means for testing H : τ = τ = L = τ 0 1-1,b-b-+1

We will reject when H : τ = τ = L = τ 0 1 mst >F-1,b--b+1,α mse The results of a RCBD Analysis of Variance can be summarized and displayed in an ANOVA Table: Source of Variation Degrees of Freedom (df) Sum of Squares (SS) Mean Square (MS) F-Ratio Block b-1 Treatment -1 Error b b +1 b ( h ) h=1 ( ) i i=1 sst = b y - y ssθ = y - y sse = sstot - ssθ -sst ssb msb = b-1 sst mst = -1 mst F= mse sse mse = b -b- +1 Total n - 1 ( b hi ) sstot = y - y h=1 i=1

Example Again consider the original One-Factor CRD Automobile Battery Life problem with the addition of information regarding the brand of battery (TryHard, NeverStart, WorryLast, or Fail-Safe): Automobile Battery Life (in Months) Brand New England Mid West South West TryHard 48 51 39 NeverStart 4 45 43 WorryLast 43 56 38 Fail-Safe 47 5 36 Region is the treatment (i = 1 for New England, i = for Mid West, and i = 3 for South West) and Brand is the block (h = 1 for TryHard, h = for NeverStart, h = 3 for WorryLast, and h = 4 for Fail-Safe). Use these data to perform a RCBD ANOVA. Here s the interaction plot for this data: Life (in Months) Interaction Plot 60 50 40 30 0 10 Sub-Compact TryHard Compact NeverStart Midsized WorryLast Full Fail-Safe Sized 0 New TryHard England NeverStart Mid West WorryLast South West Brand Region There is very little evidence of interaction between regions and brand of battery (the relationships between regions and life of battery are similar for the different brands).

Or this interaction plot: Life (in Months) Interaction Plot 60 50 40 30 0 10 New TryHard England Mid NeverStart West South WorryLast West 0 TryHard Sub- Compact NeverStart Compact Midsized WorryLast Fail-Safe Full Sized Brand Again, there is only moderate evidence of interaction between region and brand of battery. This could be used to make us comfortable in not interacting the treatment and block. So we proceed with a test of the main effect associated with region: H0 : τ NewEngland = τ MidWest = τsouthwest The appropriate sum of squares is ( ) y i y sst = b y i - y = - = 4588-4300 = 88 b b i=1 i=1 and the corresponding mean square is sst 88 mst = = = 144.0-1

To calculate the F-statistic necessary to test the region effect hypothesis, we still need the sse: sse = sstot - ssθ - sst for which we need the sstot and ssθ to calculate. From our previous work we already know that sstot = 40.0, so if we find ssθ we can easily calculate sse. The ssθ for the RCBD main effects model is y y ssθ = - b = b h h=1 ( 48 + 4 + 43 + 47 ) + ( 51 + 45 + 56 + 5 ) + ( 39 + 43 + 38 + 36) ( 48 + 4 + 43 + 47 + 51 + 45 + 56 + 5 + 39 + 43 + 38 + 36) - 1 = 431.67-4300 = 1.67 so we have that sse = sstot - ssθ - sst = 40-88 1.67 = 101.33 and the corresponding mean squares is sse 101.33 mse = = = 16.8888 b -b- +1 6 3

which results in an F-statistic for the hypothesis test of a region main effect of mst 144 F = = = 8.566 mse 16.8888 At 1 = and b b - + 1 = 6 degrees of freedom, this test statistic has a p-value of 0.01763, so we have fair evidence that the hypothesis of no treatment (region) effect is incorrect. The results of our RCBD Analysis of Variance can be summarized and displayed in an ANOVA Table: Source of Variation Degrees of Freedom (df) Sum of Squares (SS) Mean Square (MS) F-Ratio Block 3 ssθ = 1.67 msθ = 4.3 Treatment sst = 88.0 mst = 144.0 F = 8.5 Error 6 sse = 101.33 mse = 16.88 Total 11 sstot = 40.0

SAS code for an RCBD Model: DATA battery; INPUT region $ autotype $ trtmnt ion life brand $; num=_n_; LABEL region= Region of Country' autotype='type of Auto (Mid- vs. Full-sized)' life = 'Observed Life of Tire in Months' ion = 'Received Ion Treament (yes=1, no=0)' numdays = 'Number of Days with High Temperature Below Freezing' brand= Brand of Battery' num = 'Order of Data'; CARDS; NewEngla Mid 1 1 48 36 TryHard......... SouthWes Full 6 0 36 45 Fail-Safe ; PROC GLM DATA=battery; TITLE4 'Using PROC GLM for an RCBD Analysis'; CLASS brand region; MODEL life=brand region; RUN; SAS output for an RCBD Model: Analysis of Variance AUTOMOBILE BATTERY LIFE DATA ANALYSIS FOR QA 605 WINTER QUARTER 006 Using PROC GLM for an RCBD Analysis Dependent Variable: life The GLM Procedure Observed Life of Battery in Months Sum of Source DF Squares Mean Square F Value Pr > F Model 5 300.6666667 60.1333333 3.56 0.0769 Error 6 101.3333333 16.8888889 Corrected Total 11 40.0000000 R-Square Coeff Var Root MSE life Mean 0.74797 9.13465 4.109609 45.00000 Source DF Type I SS Mean Square F Value Pr > F brand 3 1.6666667 4. 0.5 0.8587 region 88.0000000 144.0000000 8.53 0.0176 Source DF Type III SS Mean Square F Value Pr > F brand 3 1.6666667 4. 0.5 0.8587 region 88.0000000 144.0000000 8.53 0.0176

- Multiple comparisons and the RCBD the blocktreatment model is similar to a single replicate two-way main effects model, so the least squares estimator for each µ + θ h + τ i (h = 1,,b; i = 1,,) is similar to the least squares estimator for each µ + α i + β j (i = 1,,a; j = 1,,b) so analogously any contrast in the treatment effects c τ, c = 0 i i i=1 i=1 is estimable in the randomized complete block design and has least-squares estimator i c ˆτ = c Y i i i i i=1 i=1 This contrast also has variance ˆ ci Var ciτ i = Var ciy i = σ i=1 i=1 i=1 b which leads to a 100(1 α)% confidence interval for a treatment contrast of c ˆ ± ˆ iτ i ciτ i t α Var b-b-+1, cτ i i i=1 i=1 i=1 and multiple comparisons of the form ciτ i cy i i± w mse c i b i=1 i=1 i=1

For various multiple comparison procedures, we have the following critical coefficients: w B = t b-b- +1, α m ( ) w S = -1 F q w T =,b-b-+1,α -1,b-b- +1,α (where q, b -b + 1, α is taken from Table A.8 in D&V) w = t D (where ( 0.5) -1,b-b- +1,α ( 0.5) t -1,b-b-+1,α is taken from Table A.10 in D&V) ( 0.5) H D1-1,b-b- +1,α ( 0.5) t -1,b-b- +1,α w = w = t (where is taken from Table A.9 in D&V) D.The General Complete Block Design the Block-Treatment Model) We have treatments (which may be factorial treatment combinations) and b blocks of k > (relatively) homogenous within and heterogeneous between groups (so s > 1 and r > b) Now let A be a factor with levels i = 1,,a B be a blocking factor with levels h = 1,,b s be the number of replicates Y hit = the response obtained observation from the h th block randomly assigned of the i th treatment hit = the error variable (or variation attributable to randomness) The basic model is now Y = µ + θ + τ + hit h i hit

We again need to make some assumptions about the error terms hit in order to statistically analyze the results of a RCBD experiment: - hit ~ N(0, σ ) - hit s are mutually independent, t=1,,s, h = 1,,b, i = 1,, We can test these assumptions using techniques discussed earlier. This model is similar to the previous treatment-block model except that we have more than one replicate. So we have a single standard hypothesis: H : τ = τ = L = τ 0 1 i.e., there is no treatment effect. Note that this can be partitioned into hypothesis tests about various factors in more complex designs. To test this hypothesis, we modify our previous formulas for the various sums of squares to account for multiple replicates. - the sum of squares for treatments is y sst = bs y - y = - bs ( ) i i i=1 i=1 - the sums of squares for blocks is y ssθ = s y - y = - s y bs b b ( ) h h h=1 h=1 y bs

- the total sum of squares is y sstot = y - y = y - bs b s b s ( hit ) hit h=1 i=1 t=1 h=1 i=1 t=1 - these sums of squares can be used to find the sum of squares for error: sse = sstot - ssθ - sst Now if H 0 is true, then sst ~ χ σ -1 sse and since ~ χ bs-b-+1 σ and sst and sse are independent, the ratio of these two Chi-Square statistics (divided by their respective degrees of freedom) yields sst sst ( -1) σ ( -1) mst = = ~ F sse sse mse ( bs-b- +1) σ ( bs-b- +1) which provides us with the means for testing H : τ = τ = L = τ 0 1-1,bs-b-+1

We will reject when H : τ = τ = L = τ 0 1 mst >F-1,b--b+1,α mse The results of an Analysis of Variance for a RCBD with replicates can be summarized and displayed in an ANOVA Table: Source of Variation Degrees of Freedom (df) Sum of Squares (SS) Mean Square (MS) F-Ratio Block b-1 Treatment -1 Error bs b +1 b ( h ) h=1 ( ) i i=1 sst = bs y - y ssθ = s y - y sse = sstot - ssθ -sst ssb msb = b-1 sst mst = -1 mst F= mse sse mse = bs-b- +1 Total n - 1 b s ( hit ) sstot = y - y h=1 i=1 t=1

Example Now consider the One-Factor CRD Automobile Battery Life problem where we wish to consider type of battery (sealed or non-sealed) as a blocking factor: Automobile Battery Life (in Months) Sealed? New England Mid West South West No 4 45 43 No 47 5 36 Yes 48 51 39 Yes 43 56 38 Region is the treatment (i = 1 for New England, i = for Mid West, and i = 3 for South West) and Type of Battery is the block (j = 1 for Sealed and j = for Nonsealed). Use these data to perform a General RCBD ANOVA. Here s the interaction plot for this data: Life of Battery (Months) 60 Interaction Plot 50 40 30 0 Sealed Nonsealed 10 0 New TryHard England NeverStart Mid West WorryLast South West Region Brand There is very little evidence of interaction between region and type of battery (the relationships between regions and life of battery are similar for the different types of battery).

Or this interaction plot: Life of Battery (Months) 60 Interaction Plot 50 40 30 0 TryHard New England NeverStart Mid West WorryLast South West 10 0 Sealed Nonsealed Type of Battery Again, there is little evidence of interaction between region and type of battery. This could be used to make us comfortable in not interacting the treatment and block. So we proceed with a test the main effect associated with region: H0 : τ NewEngland = τ MidWest = τsouthwest The appropriate sum of squares is ( ) y i y sst = bs y i - y = - = 4588-4300 = 88 bs bs i=1 i=1 and the corresponding mean square is sst 88 mst = = = 144.0-1

To calculate the F-statistic necessary to test the region effect hypothesis, we still need the sse: sse = sstot - ssθ - sst for which we need the sstot and ssθ to calculate. From our previous work we already know that sstot = 40.0, so if we find ssθ we can easily calculate sse. The ssθ for the RCBD main effects model is y y ssθ = - = 4308.33-4300 = 8.33 bs b h h=1 s so we have that sse = sstot - ssθ - sst = 40-88 8.33 = 105.67 and the corresponding mean squares is sse 105.67 mse = = = 13.0875 bs-b- +1 8 which results in an F-statistic for the hypothesis test of a region main effect of mst 144 F = = = 10.90186 mse 13.1 At 1 = and bs b - + 1 = 8 degrees of freedom, this test statistic has a p-value of 0.005191, so we have strong evidence that the hypothesis of no treatment (region) effect is incorrect.

The results of our General RCBD Analysis of Variance can be summarized and displayed in an ANOVA Table: Source of Variation Degrees of Freedom (df) Sum of Squares (SS) Mean Square (MS) F-Ratio Block 1 ssθ = 8.33 msθ = 8.33 Treatment sst = 88.0 mst = 144.0 F = 10.90 Error 8 sse = 105.67 mse = 13.1 Total 11 sstot = 40.0 SAS code for an RCBD Model: DATA battery; INPUT region $ autotype $ trtmnt ion life numdays brand $; num=n_; LABEL region= Region of Country' autotype='type of Auto (Mid- vs. Full-sized)' life = 'Observed Life of Battery in Months' ion = 'Received Ion Treament (yes=1, no=0)' numdays = 'Number of Days with High Temperature Below Freezing' brand= Brand of Battery' num = 'Order of Data'; CARDS; NewEngla Mid 1 1 48 36 TryHard......... SouthWes Full 6 0 36 45 Fail-Safe ; PROC GLM DATA=battery; TITLE4 'Using PROC GLM for a General RCBD Analysis with Replicates'; CLASS type region; MODEL life=type region; RUN;

SAS output for an RCBD Model: Analysis of Variance AUTOMOBILE BATTERY LIFE DATA ANALYSIS FOR QA 605 WINTER QUARTER 006 Using PROC GLM for a General RCBD Analysis with Replicates Dependent Variable: life The GLM Procedure Observed Life of Battery in Months Sum of Source DF Squares Mean Square F Value Pr > F Model 3 96.3333333 98.7777778 7.48 0.0104 Error 8 105.6666667 13.083333 Corrected Total 11 40.0000000 R-Square Coeff Var Root MSE life Mean 0.737148 8.0768 3.63437 45.00000 Source DF Type I SS Mean Square F Value Pr > F type 1 8.3333333 8.3333333 0.63 0.4499 region 88.0000000 144.0000000 10.90 0.005 Source DF Type III SS Mean Square F Value Pr > F type 1 8.3333333 8.3333333 0.63 0.4499 region 88.0000000 144.0000000 10.90 0.005 E. The General Complete Block Design the Block-Treatment Interaction Model) We have treatments (which may be factorial treatment combinations) and b blocks of k > (relatively) homogenous within and heterogeneous between groups (so s > 1 and r > b) Now let A be a factor with levels i = 1,,a B be a blocking factor with levels h = 1,,b s be the number of replicates Y hit = the response obtained observation from the h th block randomly assigned of the i th treatment hit = the error variable (or variation attributable to randomness) The basic model is now ( ) Y = µ + θ + τ + θτ + hit h i hi hit

We again need to make some assumptions about the error terms hit in order to statistically analyze the results of a RCBD experiment: - hit ~ N(0, σ ) - hit s are mutually independent, t=1,,s, h = 1,,b, i = 1,, We can test these assumptions using techniques discussed earlier. This model is similar to the previous treatment-block with replicates model except that we now allow for a block-treatment interaction. Our first standard hypothesis for this model is: ( ) ( ) ( ) ( ) H : θτ - θτ - θτ + θτ = 0 i,j θt 0 hi h i i.e., there is no block-treatment interaction. To test this hypothesis, we modify our previous formulas for the various sums of squares to account for the blocktreatment interaction. - the sum of squares for treatments is unchanged: y sst = bs y - y = - bs ( ) i i i=1 i=1 y bs - the sums of squares for blocks is also unchanged: y ssθ = s y - y = - s b b ( ) h h h=1 h=1 y bs

- Of course, the total sum of squares is unchanged: y sstot = y - y = y - bs b s b s ( hit ) hit h=1 i=1 t=1 h=1 i=1 t=1 - these sums of squares can be used to find the sum of squares for error: sse = sstot - ssθ - sst - ssθt which can not be calculated in this manner until we find the sum of squares for the block-treatment interaction. - the sum of squares for the block-treatment interaction is: b b b yhi y i yh y ssθt =s ( yhi -y ) = - - + s bs s bs h=1 i=1 h=1 i=1 i=1 h=1 Now we can find the sum of squares for error: sse = sstot - ssθ - sst - ssθt

ssθt Now if the null hypothesis is true, then ~ χ ( b-1)( -1) σ sse and since ~ χ b( s-1) σ and ssθt and sse are independent, the ratio of these two Chi-Square statistics (divided by their respective degrees of freedom) yields ssθt ssθt ( b-1) ( -1) σ ( b-1) ( -1) msθt = = ~ F sse sse, mse b ( s-1) σ b ( s-1) which provides us with the means for testing ( ) ( ) ( ) ( ) θt 0 hi h i ( b-1 )( -1) b ( s-1 ) H : θτ - θτ - θτ + θτ = 0 i,j We will reject ( ) ( ) ( ) ( ) H : θτ - θτ - θτ + θτ = 0 i,j θt 0 hi h i when mst >Fb-1 ( )( -1 ), b ( s-1 ) mse,α At this point we can proceed with the second standard hypothesis for this model: T H : τ = τ = L = τ 0 1 i.e., there is no treatment effect. Again note that this can be partitioned into hypothesis tests about various factors in more complex designs.

sst Now if the null hypothesis is true, then ~ χ -1 σ sse and since ~ χ b( s-1) σ and sst and sse are independent, the ratio of these two Chi-Square statistics (divided by their respective degrees of freedom) yields sst sst ( -1) σ ( -1) mst = = ~ F sse sse -1,b ( s-1 ) mse b ( s-1) σ b ( s-1) which provides us with the means for testing H : τ = τ = L = τ 0 1 We will reject when H : τ = τ = L = τ 0 1 mst >F-1,b ( s-1 ) mse,α

The results of an Analysis of Variance for a RCBD with replicates can be summarized and displayed in an ANOVA Table: Source of Variation Degrees of Freedom (df) Sum of Squares (SS) Mean Square (MS) F-Ratio Block b-1 Treatment -1 Interaction(b-1)( 1) Error b(s-1) Total n - 1 sst = bs b ( h ) ssθ = s y - y h=1 ( y - y ) i i=1 b ( hi ) ssθt =s y -y h=1 i=1 sse = sstot - ssθ- sst - ssθt b s ( hit ) sstot = y - y h=1 i=1 t=1 ssb msb = b-1 sst mst = -1 ssθt msθt = ( b-1 ) ( -1 ) sse mse = b s-1 ( ) mst F= mse msθt F= mse Example Again consider the One-Factor CRD Automobile Battery Life problem where we wish to consider whether the battery is sealed as a blocking factor: Automobile Battery Life (in Months) Sealed? New England Mid West South West No 4 45 43 No 47 5 36 Yes 48 51 39 Yes 43 56 38 Region is the treatment (i = 1 for New England, i = for Mid West, and i = 3 for South West) and Type of Battery is the block (j = 1 for Sealed and j = for Nonsealed). Use these data to perform a General Complete RCBD ANOVA with a block-treatment interaction.

First we test for a block-treatment interaction: ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 ( ) ( ) ( ) ( ) 13 1 3 ( ) ( ) ( ) ( ) 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) H : θτ - θτ - θτ + θτ = 0, θt 0 11 1 1 θτ - θτ - θτ + θτ = 0, θτ - θτ - θτ + θτ = 0, θτ - θτ - θτ + θτ = 0, θτ - θτ - θτ + θτ = 0, θτ - θτ - θτ + θτ = 0, 3 3 The appropriate sum of squares is b b yhi y i yh y ssθt = - - + h=1 i=1 s i=1 bs h=1 s bs = 4615-4588 - 4308.33 + 4300 = 18.67 and the corresponding mean square is ssθt 18.67 msθt = = = 9.33 ( b-1) ( -1) ( -1) ( 3-1) The revised sum of squares for error is sse = sstot - ssθ - sst - ssθt = 40-88 8.33 18.67 = 87.00 and the corresponding mean squares is sse 87.00 mse = = = 14.50 b s-1-1 ( ) ( )

The resulting F-statistic for testing the hypothesis no block-treatment interaction is msθt 9.33 F = = = 0.64368 mse 14.50 At (b 1)( 1) = and b(s 1) = 6 degrees of freedom, this test statistic has a p-value of 0.558141, so we have very weak evidence of the existence of a brand-treatment interaction. Now we execute the test for a treatment effect: H0 : τ TryHard = τ NeverStart = τworrylast We have already calculated the sum of squares for treatment to be ( ) y i y sst = bs y i - y = - = 4588-4300 = 88 bs bs i=1 i=1 and the corresponding mean square is sst 88 mst = = = 144.0-1

The resulting F-statistic for testing the hypothesis no treatment effect is mst 144.00 F = = = 9.93103 mse 14.50 At (b 1)( 1) = and b(s 1) = 6 degrees of freedom, this test statistic has a p-value of 0.01487, so we have fairly strong evidence of the existence of a treatment effect. The results of our Analysis of Variance for a RCBD with replicates can be summarized and displayed in an ANOVA Table: Source of Variation Degrees of Freedom (df) Sum of Squares (SS) Mean Square (MS) F-Ratio Block 1 ssθ = 8.33 msb = 8.33 Treatment mst = 144.00 F = 9.93 sst=88 Interaction ssθt = 18.67 msθt = 9.33 F = 0.64 Error 6 sse = 87.0 mse = 14.50 Total 11 sstot = 40.0

SAS code for a RCBD Model with a Block-Treatment Interaction: DATA battery; INPUT region $ autotype $ trtmnt ion life numdays brand $; num=_n_; LABEL region= Region of Country' autotype='type of Auto (Mid- vs. Full-sized)' life = 'Observed Life of Battery in Months' ion = 'Received Ion Treament (yes=1, no=0)' numdays = 'Number of Days with High Temperature Below Freezing' brand= Brand of Battery' num = 'Order of Data'; CARDS; NewEngla Mid 1 1 48 36 TryHard......... SouthWes Full 6 0 36 45 Fail-Safe ; PROC GLM DATA=battery; TITLE4 'Using PROC GLM for a General Complete RCBD Analysis'; CLASS type region; MODEL life=type region; RUN; SAS output for a RCBD Model with a Block-Treatment Interaction: Analysis of Variance AUTOMOBILE BATTERY LIFE DATA ANALYSIS FOR QA 605 WINTER QUARTER 006 Using PROC GLM for a General Complete RCBD Analysis Dependent Variable: life The GLM Procedure Observed Life of Battery in Months Sum of Source DF Squares Mean Square F Value Pr > F Model 5 315.0000000 63.0000000 4.34 0.0510 Error 6 87.0000000 14.5000000 Corrected Total 11 40.0000000 R-Square Coeff Var Root MSE life Mean 0.78358 8.461970 3.807887 45.00000 Source DF Type I SS Mean Square F Value Pr > F type 1 8.3333333 8.3333333 0.57 0.4771 region 88.0000000 144.0000000 9.93 0.015 type*region 18.6666667 9.3333333 0.64 0.5581 Source DF Type III SS Mean Square F Value Pr > F type 1 8.3333333 8.3333333 0.57 0.4771 region 88.0000000 144.0000000 9.93 0.015 type*region 18.6666667 9.3333333 0.64 0.5581

- Multiple comparisons and the General CBD again, the block-treatment model is similar to a single replicate two-way main effects model, so the least squares estimator for each µ + θ h + τ i (h = 1,,b; i = 1,,) is similar to the least squares estimator for each µ + α i + β j (i = 1,,a; j = 1,,b) so analogously any contrast in the treatment effects c τ, c = 0 i i i=1 i=1 is estimable in the randomized complete block design and has least-squares estimator i c ˆτ = c Y i i i i i=1 i=1 This contrast also has variance ˆ ci Var ciτ i = Var ciy i = σ i=1 i=1 i=1 b which leads to a 100(1 α)% confidence interval for a treatment contrast of c ˆ ± ˆ iτ i ciτ i t α Var n-b--1, cτ i i i=1 i=1 i=1 and multiple comparisons of the form ciτ i cy i i± w mse c i b i=1 i=1 i=1

For various multiple comparison procedures, we have the following critical coefficients: w B = t n-b- +1, α m ( ) w S = -1 F q,n- b- +1,α w T = -1,n-b- +1,α (where q, n b - + 1, α is taken from Table A.8 in D&V) w = t D (where ( 0.5) -1,n-b- +1,α ( 0.5) t -1,n-b- +1,α is taken from Table A.10 in D&V) ( 0.5) H D1-1,n-b- +1,α ( 0.5) t -1,n-b- +1,α w = w = t (where is taken from Table A.9 in D&V) The block-treatment interaction model is analogous to the two-way complete model, and contrasts and multiple comparison tests (for both the block-treatment interaction and treatment effect) are handled in a similar manner.

F. Sample Size Calculations Recall that a complete block design has n = bs experimental units divided into b blocks of size k = s. Thus block size k must be chosen to accommodate the experimental conditions budget constraints desired power/width of confidence intervals Thus, computing the number of observations s per block that will achieve a given level of power of the test of no treatment differences is analogous sample size calculations for testing main effects in a two-way design: s σφ b G.Checking Model Assumptions and Responding to their Violations Assumptions on block models (block-treatment and block-treatment interaction models) must be checked in the usual manner this can be accomplished by plotting residuals against order of observation (for independence) predicted values y hit, levels of treatment factor, levels of block factor (homogeneity of variance and fit/outliers) normal scores also against each treatment (or block) if r (or k) is large (normality of residuals) Of course, other approaches to assessment of assumptions (HOV tests, Shapiro-Wilk test, etc.) should also be used where appropriate.

If the assumptions are not met, we have similar alternatives transform the response to normalize the distribution and/or stabilize the variance of the residuals Brown-Forsythe Modified F-Test (under nonconstant variance) Satterthwaite s approximation or Welch s test (for multiple comparisons) a nonparametric approach, i.e., Friedman s test for randomized complete block designs (under nonconstant variance and/or nonnormal residuals) Friedman s Rank Test for Differences in c Medians - a common distribution-free alternative to ANOVA for RCBDs. Friedman s Test considers the following hypothesis of medians (M i s): H 0 : M 1 = =M To use Friedman s test 1. Rank all observations in descending order within each block. Calculate Friedman s test statistic F R : 1 s F R = T i - 3k ( s + 1) ks s + 1 ( ) i=1 where T i is the sum of ranks associated with the i th treatment. Note that if H 0 is true (no differences in treatment medians exist) then H ~ χ k-1.

Example Again consider the original One-Factor CRD Automobile Battery Life problem with the addition of information regarding the brand of battery (TryHard, NeverStart, WorryLast, or Fail-Safe): Automobile Battery Life (in Months) Brand New England Mid West South West TryHard 48 51 39 NeverStart 4 45 43 WorryLast 43 56 38 Fail-Safe 47 5 36 Region is the treatment (i = 1 for New England, i = for Mid West, and i = 3 for South West) and Brand is the block (h = 1 for TryHard, h = for NeverStart, h = 3 for WorryLast, and h = 4 for Fail-Safe). Suppose we believe Brand and Type of Automobile do not interact. Use these data to perform Friedman s Test. First we rank all observations in descending order within each block: Automobile Battery Life (in Months) Brand New England Mid West South West TryHard 1 3 NeverStart 3 1 WorryLast 1 3 Fail-Safe 1 3 Note that the sums of the ranks for the treatments are T NewEngland = 9, T MidWest = 4, and T SouthWest =11.

Now calculate Friedman s test statistic F R : 1 F R = T - 3k s + 1 ks s + 1 ( ) s i i=1 ( ) 1 = 9 + 4 + 11-3 4 3 + 1 = 6.50 4 3 3 + 1 ( ) ( ) ( ) ( ) ( ) At k 1 = 3 degrees of freedom, the computed value of the test statistic has a p-value of 0.038774, so we have some evidence against the null hypothesis New England = MidWest = SouthWest M M M SAS code for Friedman s test: DATA battery; INPUT region $ autotype $ trtmnt ion life ion numdays brand $; LABEL region= Region of Country' autotype='type of Auto (Mid- vs. Full-sized)' life = 'Observed Life of Battery in Months' ion = 'Received Ion Treament (yes=1, no=0)' numdays = 'Number of Days with High Temperature Below Freezing' brand= Brand of Battery'; CARDS; NewEngla Mid 1 1 48 36 TryHard...... SouthWes Full 6 0 36 45 Fail-Safe ; PROC SORT battery; BY size; RUN; PROC RANK DATA=battery OUT=rbattery; VAR life; BY brand; ranks rlife; run; proc freq data=rbattery; tables brand*region*rlife/noprint cmh; run;

SAS output for Friedman s test: Using PROC FREQ for Friedman s Test The FREQ Procedure Summary Statistics for region by rlife Controlling for brand Cochran-Mantel-Haenszel Statistics (Based on Table Scores) Statistic Alternative Hypothesis DF Value Prob ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 1 Nonzero Correlation 1 6.150 0.0133 Row Mean Scores Differ 6.5000 0.0388 3 General Association 4 10.0000 0.0404 Total Sample Size = 1 H.Factorial Experiments and Blocking When treatments are factorial in nature, treatment parameter τ i in the complete block model can be replaced by the associated main effect and interaction parameters For an experiment with two interacting treatment factors in a randomized complete block design, the treatment-block model is Y = µ + θ + τ + hijt h ij hijt substitution of the main effect and interaction parameters for the treatment parameters τ ij results in the equivalent model ( ) Y = µ + θ + γ + δ + γδ + hijt h i j ij hijt This can be generalized to any number of treatment and/or blocking factors.