QNT 531 Advanced Problems in Statistics and Research Methods

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QNT 531Advanced Problems in Statistics and
Research Methods

• WORKSHOP 2
• By
• Dr. Serhat Eren
• University OF PHOENIX

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SECTION 2

• ANALYSIS OF VARIANCE AND EXPERIMENTAL DESIGN

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SECTION 2SECTION OBJECTIVES

• An Introduction to Analysis of Variance
• Analysis of Variance Testing for the Equality of
  k population means
• Multiple comparison procedures
• An introduction to Experimental Design
• Completely Randomized Designs
• Randomized Block Design
• Factorial Experiment

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SECTION 2 ANALYSIS OF DATA FROM ONE-WAY DESIGNS

• One-Way Designs The Basics
• A factor is a variable that can be used to
  differentiate one group or population from
  another. It is a variable that may be related to
  he variable of interest.
• A level is one of several possible values or
  settings that the factor can assume.
• The response variable is a quantitative variable
  that you are measuring or observing.

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SECTION 2 ANALYSIS OF DATA FROM ONE-WAY DESIGNS

• These are all examples of one-way or completely
  randomized designs.
• An experiment has a one-way or completely
  randomized design if there are several different
  levels of one factor being studied and the
  objects or people being observed/ measured are
  randomly assigned to one of the levels of the
  factor.

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SECTION 2 ANALYSIS OF DATA FROM ONE-WAY DESIGNS

• The term one-way refers to the fact that the
  groups differ with regard to the one factor being
  studied.
• The term completely randomized refers to the fact
  that individual observations are assigned to the
  groups in a random manner.

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SECTION 2 ANALYSIS OF DATA FROM ONE-WAY DESIGNS

• Understanding the Total Variation
• Analysis of variance (ANOYA) is the technique
  used to analyze the variation in the data to
  determine if more than two population means are
  equal.
• A treatment is a particular setting or
  combination of settings of the factor(s)

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SECTION 2 ANALYSIS OF DATA FROM ONE-WAY DESIGNS

• The grand mean or the overall mean is the sample
  average of all the observations in the
  experiment. It is labeled (x-bar-bar).
• Now we can rewrite the variance calculations as
  follows

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SECTION 2 ANALYSIS OF DATA FROM ONE-WAY DESIGNS

• The total variation or sum of squares total (SST)
  is a measure of the variability in the entire
  data set considered as a whole.
• SST is calculated as follows

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SECTION 2 ANALYSIS OF DATA FROM ONE-WAY DESIGNS

• Components of Total Variation
• The between groups variation is also called the
  Sum or Squares between or the Sum of Squares
  Among and it measures how much of the total
  variation comes from actual differences in the
  treatments.
• The dot-plot shown in Figure 14.3 displays the
  sample average for each of the four time
  treatments. These are called treatment means.

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SECTION 2 ANALYSIS OF DATA FROM ONE-WAY DESIGNS

• A treatment mean is the average of the response
  variable for a particular treatment.
• Between Groups Variation measures how different
  the individual treatment means are from the
  overall grand mean. It is often called the sum of
  squares between or the sum of squares among (SSA).

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SECTION 2 ANALYSIS OF DATA FROM ONE-WAY DESIGNS

• The formula for sum of squares among (SSA) is
• Within groups variation measures the variability
  in the measurements within the groups. It is
  often called sum of squares within or sum of
  squares error (SSE).

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SECTION 2 ANALYSIS OF DATA FROM ONE-WAY DESIGNS

• The Mean Square Terms in the ANOVA Table
• The mean square among is labeled MSA The mean
  square error is labeled MSE and the mean square
  total is labeled MST.
• The formulas for the mean squares are

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SECTION 2 ANALYSIS OF DATA FROM ONE-WAY DESIGNS

• Testing the Hypothesis of Equal Means
• In general, the null and alternative hypotheses
  for a one-way designed experiment are shown
  below
• HA At least one of the population means is
  different from the others.

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SECTION 2 ANALYSIS OF DATA FROM ONE-WAY DESIGNS

• The formula for the F test statistic is
  calculated by taking the ratio of the two sample
  variances
• In ANOVA, MSA and MSE are our two sample
  variances. So the F statistic is calculated as

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SECTION 2 ASSUMPTIONS OF ANOVA

• The three major assumptions of ANOVA are as
  follows
• The errors are random and independent of each
  other.
• Each population has a normal distribution.
• All of the populations have the same variance.

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SECTION 2ANALYSIS OF DATA FROM BLOCKED DESIGNS

• A block is a group or objects or people that have
  been matched. Are object or person can be matched
  with itself, meaning that repeated observations
  are taken on that object or person and these
  observations form a block?
• If the realities of data collection lead you to
  use blocks, then you must take this into account
  in your analysis. Your experimental design is
  called a randomized block design. Instead of
  using a one-way ANOVA you must use a block ANOVA.

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SECTION 2ANALYSIS OF DATA FROM BLOCKED DESIGNS

• An experiment has a randomized block design if
  several different levels of one factor are being
  studied and the objects or people being observed/
  measured have been matched.
• Each object or person is randomly assigned to one
  of the c levels of the factor.

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SECTION 2ANALYSIS OF DATA FROM BLOCKED DESIGNS

• Partitioning the Total Variation
• Like the approach we took with data from a
  one-way design, the idea is to take the total
  variability as measured by SST and break it down
  into its components.
• With a block design there is one additional
  component the variability between the blocks. It
  is called the sum of squares blocks and is
  labeled SSBL.

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SECTION 2ANALYSIS OF DATA FROM BLOCKED DESIGNS

• The sum of squares blocks measures the
  variability between the blocks. It is labeled
  SSBL.
• For a block design, the variation we see in the
  data is due to one of three things the level of
  the factor, the block, or the error.

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SECTION 2ANALYSIS OF DATA FROM BLOCKED DESIGNS

• Thus, the total variation is divided into three
  components
• SST SSA SSBL SSE

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SECTION 2ANALYSIS OF DATA FROM BLOCKED DESIGNS

• Using the ANOVA Table in a Block Design
• The ANOVA table for such a block design looks
  just like the ANOVA table for a one-way design
  with an additional row.

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SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS

• Motivation for a Factorial Design Model
• An experimental design is called a factorial
  design with two factors if there are several
  different levels of two factors being studied.
• The first factor is called factor A and there are
  r levels of factor A. The second factor is called
  factor B and there are c levels of factor B.

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SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS

• The design is said to have equal replication if
  the same number of objects or people being
  observed/measured are randomly selected from each
  population.
• The population is described by a specific level
  for each of the two factors. Each observation is
  called a replicate.
• There are n' observations or replicates observed
  from each population. There are n n'rc
  observations in total.

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SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS

• Partitioning the Variation
• The sum of squares due to factor A is labeled
  SSA. It measures the squared differences between
  the mean of each level of factor A and the grand
  mean.
• The sum of squares due to factor B is labeled
  SSB. It measures the squared differences between
  the mean of each level of factor B and the grand
  mean.

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SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS

• The sum of squares due to the interacting effect
  of A and B is labeled SSAB. It measures the
  effect of combining factor A and factor B.
• The sum of squares error is labeled SSE. It
  measures the variability in the measurements
  within the groups.
• Thus, the total variation is divided into four
  components
• SST SSA SSB SSAB SSE

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SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS

• Using the ANOVA Table in a Two-Way Design
• The ANOVA table for such a design looks just like
  the ANOVA table for a one-way design with two
  additional rows.

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SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS

• Using the ANOVA Table in a Two-Way Design
• In a two-way ANOVA, three hypothesis tests should
  be done.
• To test the hypothesis of no difference due to
  factor A we would have the following null and
  alternative hypotheses
• Ho There is no difference in the population
  means due to factor A.
• HA There is a difference in the population
  means due to factor A.

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SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS

• To test the hypothesis of no difference due to
  factor B we would have the following null and
  alternative hypotheses
• Ho There is no difference in the population
  means due to factor B.
• HA There is a difference in the population
  means due to factor B.

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SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS

• To test the hypothesis of no difference due to
  the interaction of factors A and B we would have
  the following null and alternative hypotheses
• Ho There is no difference in the population
  means due to the interaction of factors A and B.
• HA There is a difference in the population
  means due to the interaction of factors A arid B.

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SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS

• Understanding the interaction Effect
• The easiest way to understand this effect is to
  look at a graph of the sample averages for each
  of the possible combinations of the two factors.
• The line graph shown in Figure 14.7 displays the
  20 sample means for airspace.

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SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS

• From this graph you can see that the mean
  airspace decreases the longer the box sits on the
  shelf, regardless of from what position in the
  hardroll the box was made.
• The airspace behavior is affected by the
  interaction of the time on the shelf and the
  position in the hardroll from which it was made.

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SECTION 2ANALYSIS OF DATA FROM TWO-WAY DESIGNS

• If there were no interaction effect, the lines
  connecting the sample means would be parallel as
  in Figure 14.8.

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SECTION 2 MULTIPLE COMPARISON PROCEDURE

• When we use analysis of variance to test whether
  the means of k populations are equal, rejection
  of the null hypothesis allows us to conclude only
  that the population means are not all equal.
• In some cases we will want to go a step further
  and determine where the differences among means
  occur.

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SECTION 2 MULTIPLE COMPARISON PROCEDURE

• The purpose of this section is to introduce two
  multiple comparison procedures that can be used
  to conduct statistical comparisons between pairs
  of population means.

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SECTION 2 MULTIPLE COMPARISON PROCEDURE

• 2.3.1 FISHERS LSD
• Suppose that analysis of variance has provided
  statistical evidence to reject the null
  hypothesis of equal population means.
• In this case, Fishers least significant
  difference (LSD) procedure can be used to
  determine where the differences occur.

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SECTION 2 MULTIPLE COMPARISON PROCEDURE

• Confidence Interval Estimate of the Difference
  Between Two Population Means Using Fishers LSD
  Procedure

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SECTION 2 MULTIPLE COMPARISON PROCEDURE

• TYPE I ERROR RATES
• We showed how Fishers LSD procedure can be used
  in such cases to determine where the differences
  occur.
• Technically, it is referred to as a protected or
  restricted LSD test because it is employed only
  if we first find a significant F value by using
  analysis of variance.

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SECTION 2 MULTIPLE COMPARISON PROCEDURE

• To see why this distinction is important in
  multiple comparison tests, we need to explain the
  difference between a comparisonwise Type I error
  rate and an experimentwise Type I error rate.

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SECTION 2 MULTIPLE COMPARISON PROCEDURE

• For example, in the NCP example Fishers LSD
  procedure was used to make three pairwise
  comparisons.

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SECTION 2 MULTIPLE COMPARISON PROCEDURE

• In each case, we used a level of significance of
  ? 0.05.
• Therefore, for each test, if the null hypothesis
  is true, the probability that we will make a Type
  I error is ? 0.05 hence, the probability that
  we will not make a Type I error on each test is
  1- 0.05 0.95.

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SECTION 2 MULTIPLE COMPARISON PROCEDURE

• In discussing multiple comparison procedures we
  refer to this probability of a Type I error
  (? 0.05) as the comparisonwise Type I error
  rate comparisonwise Type I error rates indicate
  the level of significance associated with a
  single pairwise comparison.
• Let us now consider a slightly different
  question. What is the probability that in making
  three pairwise comparisons, we will commit a Type
  I error on at least one of the three tests?

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SECTION 2 MULTIPLE COMPARISON PROCEDURE

• To answer this question, note that the
  probability that we will not make a Type I error
  on any of the three tests is
• (.95)(.95)(.95)0 .8574.
• Therefore, the probability of making at least one
  Type I error is
• 1-0.8574 0.1426

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SECTION 2 MULTIPLE COMPARISON PROCEDURE

• Thus, when we use Fishers LSD procedure to make
  all three pairwise comparisons, the Type I error
  rate associated with this approach is not .05,
  but actually 0.1426 we refer to this error rate
  as the overall or experimentwise Type I error
  rate.