Two factors in a manufacturing process for an integrated...

Two factors in a manufacturing process for an integrated circuit were studied in a two-factor experiment. The purpose of the experiment is to learn their effect on the resistivity of the wafer. The factors are implant dose ( 2 levels) and furnace position (3 levels). Experimentation is costly so only one experimental run is made at each combination. The data are as follows. $$ \begin{array}{crrr} \text { Dose } & & {\text { Position }} \\ \hline 1 & 15.5 & 14.8 & 21.3 \\ 2 & 27.2 & 24.9 & 26.1 \end{array} $$ It is to be assumed that no interaction exists between these two factors. (a) Write the model and explain terms. (b) Show the analysis of variance table. (c) Explain the 2 "error" degrees of freedom. (d) Use Tukey's test to do multiple-comparison tests on furnace position. Explain what the results show.

Two factors in a manufacturing process for an integrated circuit were studied in a two-factor experiment. The purpose of the experiment is to learn their effect on the resistivity of the wafer. The factors are implant dose ( 2 levels) and furnace position (3 levels). Experimentation is costly so only one experimental run is made at each combination. The data are as follows.
$$
\begin{array}{crrr}
\text { Dose } & & {\text { Position }} \\
\hline 1 & 15.5 & 14.8 & 21.3 \\
2 & 27.2 & 24.9 & 26.1
\end{array}
$$
It is to be assumed that no interaction exists between these two factors.
(a) Write the model and explain terms.
(b) Show the analysis of variance table.
(c) Explain the 2 "error" degrees of freedom.
(d) Use Tukey's test to do multiple-comparison tests on furnace position. Explain what the results show.

Key Concepts

Factorial Experiment Design

A factorial experiment is a design in which two or more factors are investigated simultaneously to study their effect on a response variable. Typically, each factor is tested at multiple levels, and the experiment considers every combination of these levels. This approach not only allows for the estimation of main effects of each factor but also enables the study of potential interactions between them. In the current context, one factor has 2 levels and the other has 3, resulting in 6 unique treatment combinations.

Additive Main Effects Model

When assuming no interaction between factors, the model used is called an additive or main effects model. This model expresses the response as the sum of an overall mean, a contribution from each factor's level, and a random error term. Formally, it can be written as Y_ij = ? + ?_i + ?_j + ?_ij, where ? is the overall mean, ?_i is the effect due to the i-th level of the first factor, ?_j is the effect due to the j-th level of the second factor, and ?_ij is an independent error term. This formulation simplifies the analysis by allowing the estimation of the individual factor effects without considering their interaction.

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to partition the observed variability in a response variable into components attributable to different sources, such as the factors and experimental error. In two?factor ANOVA without interaction, the total sum of squares is decomposed into components due to the first factor, the second factor, and the residual error term. An ANOVA table summarizes this decomposition along with degrees of freedom, Mean Squares, and significance tests that help determine whether the effects of the studied factors are statistically significant.

Error Degrees of Freedom

In an experimental design, degrees of freedom (df) represent the number of independent pieces of information available for estimating variability. In a factorial experiment where each treatment combination is measured only once, there is limited replication. The 'error' degrees of freedom in such a setup stem from the residual variability left after fitting the model. Even with a low number of replications, error degrees of freedom are essential for estimating the experimental noise and are used to perform F-tests in ANOVA. In the provided example with 6 observations and 1 degree of freedom for each main effect, the error degrees of freedom are minimal, reflecting the cost constraints and limited data available.

Multiple Comparisons - Tukey's Test

Tukey's test is a post-hoc multiple comparison procedure used after an overall ANOVA indicates significant differences among group means. Its purpose is to control the family-wise error rate while comparing all possible pairs of treatment means. By applying Tukey's test to the furnace position factor, the procedure examines the differences among the three mean response levels, identifying which pairs of positions differ significantly from each other. This test is particularly useful when the number of comparisons is moderate, as it provides a clear interpretation of which levels of the factor are different while maintaining a specified overall error rate.

Transcript

00:03 Right, two factors in a manufacturing process for an integrated circuit are studied in a two -factor experiment.

00:14 The purpose of the experiment is to learn their effect on the resistivity of the wafer.

00:21 The factors are implant dose, two levels, and fairness, position three levels.

00:29 Experimentation is costly, so only one experimental run is made.

00:34 At each combination.

00:37 The data is follows.

00:39 So if the dose one and two, their position, right, and the position for one, which is 15 .5, 14 .8, and 21 .3, and for two, seven, seven, two, 24 .9, and 26 .1.

00:56 It is to be assumed that no interaction exists between these two factors, write the model that's a they write the model and explain terms then be show the analysis of various table see explain the two error degrees of freedom and then do the two kiss test to to do a multiple comparison test on fairness position explain what the results show okay let's see so he says write the and explain terms so the model that are that we can come up with is y i jk right it is equals to the mu plus the alpha i plus the b j and then the alpha beta sub i j so we are saying y sub i j k is got to mu plus the alpha plus the alpha sub i j is sub i beta sub j plus the alpha beta sub i j there right plus the epsilon sub i j k there right but what are these i j and k so for i it's it's it starts from one to up to a and then j and then j j j it's close from one to up to b there right k will take the values one to up to any day.

02:45 Right.

02:45 So a takes the value.

02:47 I take the values from 1 up to a, the j values 1 up to b and then k, 1 up to.

02:57 Yeah, that's the model.

02:59 And these are the explanation for the i, j and k.

03:04 Then b says, show the analysis of variance table.

03:12 So i've prepared the analysis of variance table there.

03:16 With p values there so the computed f values are 18 .08 for the dose and position 1 .15 and the p values are for the dose is 0 .0 511 right and then for the position there is 0 .4641 there okay so that's my another table there.

03:46 Right.

03:47 Then see.

03:51 What is our c saying? our c says, explain the two error degrees of freedom.

04:00 Right.

04:01 My degree, okay, we are explaining this figure here.

04:08 Okay, so how do you get this one? we say we add these two and we separate it from five days.

04:16 Okay.

04:17 Right.

04:18 We separate it from five.

04:19 But how did you get the five the total? there are six altogether data sets there.

04:24 We separate one.

04:24 We get five there.