DESIGN OF EXPERIMENTS

DESIGN OF EXPERIMENTS

1.0 INTRODUCTION

A designed experiment is a statistical experiment and its subsequent analysis to determine if there are differences between variables acting on some outcome. It also determines which variables have the greatest impact on the outcome. An equation of the regression line may be developed so that the outcome or response can be predicted based on the variables that went into the process. The statistical technique called analysis of variance (ANOVA) is used in designed experiments to determine if there are significant differences between the variables. A good designed experiment will yield much information with relatively little data. Matrix algebra is used extensively in the analysis of experiments. There are many different types of designed experiments. Two types of experiments, the factorial design and Latin square design will be reviewed in this chapter.

2.0 FACTORIAL DESIGN

A factorial design is an arrangement in which all levels of each factor are combined with all levels of every other factor.

Factor - A variable or attribute that influences or is suspected of influencing the characteristic being investigated.

Level - The values of a factor being examined in an experiment.

Treatment - One set of levels for all factors employed in a given experiment.

Example 1

An experiment is conducted to determine the effect of temperature and pressure on the yield of a product. The numbers in the cells represent the yield (response).

	Temperature (x₁)
Pressure (x₂)	120^o	160^o
115 psi	6,8	6,10
125 psi	5,7	2,4

This is a 2² experiment. (Two factors at two levels each. The exponent indicates the number of factors). The temperature and pressure are the two factors with each having two settings or levels.

The objective of the factorial experiment is to determine:

The factors, if any, that have a significant impact on the yield

The variable or factor, if any, that has the greatest impact on the yield

The linear model (regression line) for the yield or response (Y)

2.1 The Linear Model (Regression Equation)

Y = B₀ + B₁x₁ + B₂x₂ + B₃x₁x₂ + e

Y	= yield or response
x₁	= temperature
x₂	= pressure
x₁x₂	= interaction of the two variables
e	= error or residual

B₀, B₁, B₂, and B₃ are coefficients of the variables in the equation. The equation is

Y = 6.0 - 1.5x₁ - 0.5x₂ - 1.0x₁x₂ + 1.25. The coefficients are found by matrix algebra and the various matrices that are used to compute the coefficients are shown for information on the next page. The explanation of matrix algebra techniques is beyond the scope of QReview.

2.2 Analysis of Variance (ANOVA) Table

Source	SS	d.f.	MS	F	F_Critical
Temperature (x₁)	2	1	2	0.57	7.71
Pressure (x₂)	18	1	18	5.15	7.71
Interaction (x₁x₂)	8	1	8	2.28	7.71
Residual	14	4	3.5

SS = Sum of Squares

d.f. = Degrees of Freedom

MS = Mean Square = SS/d.f. = Variance

F = F Ratio = (MS_source/MS_residual)

The critical value of F is found in the F table. Using a level of significance of .05, 1 d.f. for the numerator and 4 d.f. for the denominator, the critical value of F is 7.71. In this example the pressure has the greatest impact on the outcome, but since the F ratio for pressure does not exceed the critical F value, the impact is not statistically significant.

2.3 Coefficients for the Linear Model

Pressure (x₁)	Coded	Temperature (x₂)	Coded	Yield (y)
115	-1	120	-1	6
115	-1	160	1	6
125	1	120	-1	5
125	1	160	1	2

115	-1	120	-1	8
115	-1	160	1	10
125	1	120	-1	7
125	1	160	1	4

			x₀	x₁	x₂	x₁x₂	e
	6		1	-1	-1	1	-1
	6		1	-1	1	-1	-1
	5		1	1	-1	-1	-1
Y =	2	X =	1	1	1	1	-1
	8		1	-1	-1	1	1
	10		1	-1	1	-1	1
	7		1	1	-1	-1	1
	4		1	1	1	1	1

	1	1	1	1	1	1	1	1
	-1	-1	1	1	-1	-1	1	1
Transpose = X' =	-1	1	-1	1	-1	1	-1	1
	1	-1	-1	1	1	-1	-1	1
	-1	-1	-1	-1	1	1	1	1

	8	0	0	0	0		48
	0	8	0	0	0		-12
(X'X) =	0	0	8	0	0	(X'Y) =	-4
	0	0	0	8	0		-8
	0	0	0	0	8		10

	.125	0	0	0	0		6.0
	0	.125	0	0	0	Coefficients	-1.5
Inverse = (X'X)^-1 =	0	0	.125	0	0	B=(X'X)^-1(X'Y) =	-0.5
	0	0	0	.125	0		-1.0
	0	0	0	0	.125		1.25

Therefore B₀ = 6.0, B₁ = -1.5, B₂ = -.5, B₃ = -1.0 and e = 1.25

2.4 Calculations for the ANOVA Table

	Temperature x₁
Pressure x₂	X_1,1	X_1,2	S x₂
X_2,1	6,8	6,10	30
X_2,2	5,7	2,4	18
S x₁	26	22	48

Sx² = 330

CM is called the correction for the mean.

Let CM = (Sx)²/n_T,

CM = (48)²/8 = 288

SS_T = SS (Total) = Sx² - CM = 330 - 288 = 42

SS_t = SS (Temperature) = S(S columns)²/n_(col) - CM = [(26)² + (22)²]/4 - 288

= 290 - 288 = 2

SS_p = SS (Pressure) = S(S rows)²/n_(rows) - CM = [(30)² + (18)²]/4 - 288

= 306 - 288 = 18

SS_i = SS (Interaction) = S(S cells)²/n_(cell) - SS_t - SS_p - CM

= [(14)² + (12)² +(16)² + (6)²]/2 - 2 - 18 - 288 = 316 - 2 - 18 = 8

SS_r = SS (Residual) = [SS (Total) - SS (Temperature) - SS (Pressure) - SS (Interaction)]

= 42 - 2 - 18 - 8 = 14

2.5 Degrees of Freedom

Total: d.f. = n - 1 = 8 - 1 = 7

Temperature: d.f. = No. levels - 1 = 2 - 1 = 1

Pressure: d.f. = No. levels - 1 = 2 - 1 = 1

Interaction: d.f. = No. levels - 1 = 2 - 1 = 1

Residual: Total d.f. - S d.f. of each factor = 7 - 3 = 4

3.0 LATIN SQUARE DESIGN

A Latin square is an experimental design in which each level of each factor is combined only once with each level of two other factors or variables. A Latin square design implies the presence of three qualitative variables: rows, columns and treatments. In a Latin square experiment, it is assumed that no interactions between the variables exist.

Example 2

The following is an experiment to study the effects of four different additives on the fuel consumption of automobiles when four different drivers drive four different cars. The letters A, B, C, and D represent the additives. The additives are the treatments. The numbers in parentheses are the yields in miles per gallon. This is a 4 X 4 Latin square experiment.

	CAR
DRIVER	1	2	3	4	S x_(rows)
I	D (20)	A (21)	B (26)	C (25)	92
II	A (20)	D (23)	C (26)	B (27)	96
III	C (16)	B (15)	D (13)	A (16)	60
IV	B (20)	C (17)	A (15)	D (20)	72
S x_(columns)	76	76	80	88	320

Sx² = 6696

Sx_(TREATMENTS): A = 72, B = 88, C = 84, D = 76

3.1 The Linear Model (regression equation)

Y = Grand Mean + Driver effects + Car effects + additive effects + Residual (e)

Y = B₀ + B₁x₁ + B₂x₂ + B₃x₃ + B₄x₄ + B₅x₅ + B₆x₆ + B₇x₇ + B₈x₈ + B₉x₉ + e

Y = B₀ + driver differences + Car differences + Additive differences + e

The calculations for the coefficients of the linear model are left to the student. When the coefficients are computed and the linear model is set up, the coded values for any car, driver or additive are entered. The fuel consumption (Y) is then determined from the equation.

3.2 Analysis of Variance (ANOVA) Table

Source	SS	d.f.	MS	F	F_Critical
Cars	24	3	8	3.0	4.76
Drivers	216	3	72	27.0	4.76
Additives	40	3	13.33	5.0	4.76
Residual	16	6	2.67
Total	296	15

For a = .05, the critical F value is 4.76. Therefore, there is a significant difference between drivers and also between additives. The drivers have the greatest impact on the outcome.

3.3 Calculations for the ANOVA Table

CM = (Sx)²/n = (320)²/16 = 6400

SS_T = SS (Total) = Sx² - CM = 6696 - 6400 = 296

SS_d = SS (Drivers) = S(S Rows)²/n_(rows) - CM

= [(92)² + (96)² + (60)² + (72)²]/4 - 6400 = 6616 - 6400 = 216

SS_c = SS (Cars) = S(S Columns)²/n_(columns) - CM

=[(76)² + (76)² + (80)² + (88)²]/4 - 6400 = 6424 - 6400 = 24

SS_a = SS (Additives) = S(S treatments)²/n_(treatments) - CM

= [(72)² + (88)² + (84)² + (76)²]/4 - 6400 = 6440 - 6400 = 40

SS_r = SS (Residual or Error) = SS_T - SS_c - SS_d - SS_a = 296 - 24 - 216 - 40 = 16

3.4 Degrees of Freedom

Total: d.f. = 16 - 1 = 15

Drivers: d.f. = 4 - 1 = 3

Cars: d.f. = 4 - 1 = 3

Additives: d.f. = 4 - 1 = 3

Residual: d.f = 15 - 3 - 3 - 3 = 6

4.0 GLOSSARY OF TERMS

Analysis of variance (ANOVA) – This is a technique for determining whether there are significant differences between two or more sample averages. Usually the t test is used to test for differences between two sample averages and analysis of variance is used to test differences between three or more.

Blocked factorial – This design is used when the number of runs required for a factorial design is too large to be carried out under uniform conditions. The set of combinations of factors is divided into subsets so that some high order interactions are equated to blocks. Estimates of certain interactions are sacrificed to provide blocking. The design is the same as the factorial except that some high order interactions cannot be estimated.

Blocks – Blocks are planned groups where the experimental variability is expected to be small compared to the entire experimental layout such as a batch. Block designs are usually noise reducing such as the randomized block and Latin square designs.

Completely randomized design - This is an experiment in which all treatments are assigned to the experimental units in a completely random manner. It is appropriate when only one experimental factor is being investigated and when material is homogeneous and background conditions can be controlled. The experimental error is higher than with other designs. This results in less sensitivity.

Confounding - Confounding occurs when an effect can be attributed to two or more factors but the contribution of each cannot be determined. Confounding can also be stated as the confusing of factors so their effects cannot be separated.

Efficiency of an experimental design – The comparison of error mean squares determines the relative efficiency of an experiment. If the error mean square of a certain design is less than that of another design, using the same levels of variables, the efficiency of the first is greater than that of the second.

Experimental error – This is the variation that is not controlled in an otherwise controlled experiment. It is measured by the variance of the observations under identical conditions.

F distribution – This is a sampling distribution formed by the ratio s₁²/s₂² with n₁ – 1 and n₂ –2 degrees of freedom. The statistic s₁²is an unbiased estimate of s ₁² and s₂²is an unbiased estimate of s ₂².

F test – This is a hypothesis test to determine if two variances could have come from the same universe.

Factorial - The factorial design is used when several factors are to be investigated at two or more levels and interaction of factors are important. The design includes all observations for all combinations of levels that can be formed from the different levels.

Fractional factorial – A portion of the total combinations are selected to provide an evaluation of the main factors and the more important interactions. This design is appropriate when there are many factors and levels and it is not practical to run all combinations. Several factors are investigated at several levels but only a subset of the factorial is run.

Incomplete block design – This is a randomized block design in which not all treatments can be included in one block. It is called a balanced design if each pair of treatments occurs together the same number of times. The balanced incomplete block is used when there is one primary factor but not all treatments can be accommodated in a block. A partially balanced incomplete block design is used if a balanced incomplete block design requires a larger amount of blocks than are practical.

Interaction – Interaction means that variables are dependent on each other. In analysis of variance, it means the tendency for a combination of factors to produce a result that is different from the sum of their individual contributions.

Latin square – The Latin square is used when one primary factor is under investigation and the results may be affected by two other experimental variables. It is assumed that no interactions exist. Each treatment occurs once in every row and once in every column. The number of treatments must equal the number of rows and number of columns.

Linear model – The linear model is a regression equation referred to as a first-degree equation. It yields the expected response when various values of the independent factors or variables are input. First-degree means that the variables do not have an exponent greater than one. The independent variables are x₁, x₂, … x_n and not x₁², x₂³, … x_n^k.

Mixed model – This is an analysis of variance model in which one or more factors are fixed and one or more are random.

Nested design – This design is used when the objective is to study relative variability instead of the mean effect of sources of variation. The effects of certain factors are pyramided in this analysis of variance model. Levels of lower factors are nested in levels of factors higher up the pyramid but do not cut across the levels of the higher factor.

Noise – This is the effect of all variables, not specifically measured in the experiment, that affect the response.

Orthogonal design – An experimental design is orthogonal if the effects of one factor balance out (sum to zero) across all effects of other factors.

Randomized block design – Used when one factor is being investigated and experimental material or environment can be divided into blocks. The treatment combinations are randomized within a block and several blocks are run.

Regression equation – This is a function that indicates how the average value of the dependent variable varies with the values of the independent variable.

Replication – This is the repetition of an entire experiment. It is done to reduce errors in the experiment. Less than a complete replication would be called an incomplete block.

Residual mean square – This is the residual sum of squares divided by its degrees of freedom.

Response surface – This is a special multi-variable design for quantitative independent variables such as pressure and temperature.

Robust test – This term refers to a sample test that is not seriously affected by moderate changes from the normal pattern.

Split-plot design - One in which a main effect is confounded with blocks due to the practical necessities of the order of experimentation.

Total sum of squares – This is the sum of the squares of the deviations of the individual data value from the mean of all the data.

Treatment groups - Combinations of variables.

Two classes of designs - Noise reducing and volume shifters.

Youdin square - Same as the Latin square design but the number of rows, columns, and treatments need not be the same. Each treatment occurs once in every row. The number of treatments must equal the number of columns.