Introduction

This article will demonstrate how QLazDOE software can manage single-factor experiments with multiple levels (i.e. multiple treatments), using one-way ANOVA for analysis of results and driving conclusions.

Example used for this demonstration is from the book "Design and Analysis of Experiments" by Douglas C. Montgomery, which you can purchase from: https://www.wiley.com/en-hr/Design+and+Analysis+of+Experiments,+10th+Edition-p-9781119492443

Problem Statement

We will be dealing with a plasma etching process, where energy is supplied by a radio-frequency (RF) generator causing plasma to be generated in the gap between the electrodes. We are interested in investigating the relationship between the RF power setting and the etch rate for this tool. We want to determine whether etching rate depends on the RF power or not.  We will test four levels of RF power: 160, 180, 200, and 220 W and measure etching rate as response variable. For each power settings, there will be 5 replicates.

This is a single factor experiment, with 4 levels (i.e. 4 treatments. Results will be analyzed by using one-way ANOVA. Null hypothesis H0 is that means for 4 treatments (i.e. power settings) are equal, while alternative hypothesis is that treatment means are different.

Definition of Factors and Levels

First we need to create new project, defining its name, description and problem statement.

Now we need to create factors & levels definition file. We can do it by clicking button  or from corresponding menu item "Create New Factors and Levels Definition File" .

Design of Experiment

First we need to create new solution for the current project.

Now we will create our full factorial, replicated design of experiment, by clicking button  or choosing corresponding option "Create DOE in one step" from the menu .

This creates and opens design of experiments in a spreadsheet, ready for results data input.

Experimental Results

Next is to conduct the experiment and enter experimental results. Our response variable is etching rate, so we need to add new column "Etching Rate" and enter results.

Analysis of Experimental Results

To analyze results, we need to load results into LazStats statistical software. We can do it by clicking on the button , or by choosing corresponding menu item "Analyze DOE Results in LazStats" . This opens LazStats with loaded results. Now we have to find ANOVA function under Analyses/Comparisons/1, 2, or 3 Ways ANOVAs.

ANOVA form opens, where we need to define Etch Rate as dependent (response) variable and RF Power as input factor with fixed levels.

Click the button "Compute" in order to acquire ANOVA results.

One-Way ANOVA

LazStats returns one-way ANOVA results as follows:

ONE WAY ANALYSIS OF VARIANCE RESULTS

Dependent variable is: Etch Rate (A/min), Independent variable is: RF Power (W)

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SOURCE D.F. SS MS F PROB.>F OMEGA SQR.
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BETWEEN 3 66870.55 22290.18 66.80 0.00 0.91
WITHIN 16 5339.20 333.70
TOTAL 19 72209.75
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MEANS AND VARIABILITY OF THE DEPENDENT VARIABLE FOR LEVELS OF THE INDEPENDENT VARIABLE
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GROUP MEAN VARIANCE STD.DEV. N
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1 551.20 400.70 20.02 5
21 587.40 280.30 16.74 5
41 625.40 421.30 20.53 5
61 707.00 232.50 15.25 5
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TOTAL 617.75 3800.51 61.65 20
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TESTS FOR HOMOGENEITY OF VARIANCE
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Hartley Fmax test statistic = 1.81 with deg.s freedom: 61 and 4.
Cochran C statistic = 0.32 with deg.s freedom: 61 and 4.
Bartlett Chi-square = 0.43 with 3 D.F. Prob. > Chi-Square = 0.933
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So, our calculated F-ratio statistics, F = 66.80. Now we need to compare it with tabular value for F distribution for degrees of freedom DF between = 3, DF within = 16.

Let's go to "Statistical Tables" tab and find F Distribution sub-tab.

If we use fixed significance level of alpha = 0.05 , then we find that tabular value F0.05,3,16 = 3.24.