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ENGM 720 - Lecture 06

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ENGM 720 - Lecture 06 Multiple Comparisons, 7 Tools of Ishikawa * ENGM 720: Statistical Process Control * – PowerPoint PPT presentation

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Title: ENGM 720 - Lecture 06


1
ENGM 720 - Lecture 06
  • Multiple Comparisons,
  • 7 Tools of Ishikawa

2
Assignment
  • Reading
  • Chapter 4.5, Chapter 5 5.2, 5.4
  • Finish reading
  • Review for Exam I
  • Covers material through hypothesis tests and
    seven tools
  • Assignments
  • Obtain the Hypothesis Test (Chart ) Tables
  • Access Previous Assignment Solutions Prepare
    Notebook
  • Download Assignment 4 Assignment 4 Solutions

3
Multiple Comparisons
  • Analysis-of-variance (ANOVA) is a statistical
    method used to test hypotheses regarding more
    than two sample means.
  • For a one-factor experiment the hypothesis tested
    is

4
Multiple Comparisons
  • The strategy in an analysis of variance is to
    compare the variability between sample means to
    the variability within sample means. If they are
    the same, the null hypothesis is accepted. If the
    variability between is bigger than within, the
    null hypothesis is rejected.

Null Hypothesis
Alternative Hypothesis
5
Definitions
  • An experimental unit is the item measured during
    an experiment. The errors in these measurements
    are described by random variables.
  • It is important that the error in measurement be
    the same for all treatments (random variables be
    independent and have the same distribution).
  • The easiest way to assure the error is the same
    for all treatments is to randomly assign
    experimental units to treatment conditions.

6
Definitions
  • The variable measured in an experiment is called
    the dependent variable.
  • The variable manipulated or changed in an
    experiment is called the independent variable.
  • Independent variables are also called factors,
    and the sample means within a factor are called
    levels or treatments.

7
Definitions
  • Random samples of size n are selected from each
    of k different populations. The k different
    populations are classified on the basis of a
    single criterion or factor. (one-factor and k
    treatments)
  • It is assumed that the k populations are
    independent and normally distributed with means
    µ1, µ2, ... , µk, and a common variance s2.
  • Hypothesis to be tested is

8
Definitions
  • A fixed effects model assumes that the treatments
    have been specifically chosen by the
    experimenter, and our conclusions apply only to
    the levels chosen
  • Fixed Effect Statistical Modelwhere eij are
    independent and identically distributed N(0,s2).
  • Because the fixed effects model assumes that the
    experiment is performed in a random manner, a
    one-way ANOVA with fixed effects is often called
    a completely randomized design.

Overall Mean
ith Treatment Effect
Error in Measurement
Observed Value
9
Definitions
  • For a fixed effects model, if we restrict
  • Thenis equivalent to

10
Analysis of the Fixed Effects Model
. . .
. . .
. . .
. . .
11
Analysis of the Fixed Effects Model
  • Sum of Squares TreatmentsThe sum of squares
    treatments is a measure of the variability
    between the factor levels.
  • Sum of Squares ErrorsThe error sum of squares
    is a measure of the variability within the
    factor levels.

Sum of Squares Errors (SSE)
Factor level 1
Factor level 2
Factor level 3
X3
X1
X2
Sum of Squares Treatments (SSTr)
12
Analysis of the Fixed Effects Model
  • P-valuesThe plausibility of the null
    hypothesis (that the factor level means are all
    equal)depends upon the relative size of the
    sum of squares for treatments (SSTr) tothe sum
    of squares for errors (SSE).

13
Analysis of the Fixed Effects Model
  • Sum of Squares Partition for One Factor
    LayoutIn a one factor layout, the total
    variability in the data observations is measured
    by the total sum of squares (SST) which is
    defined to be

Total Sum of Squares SST
Treatment Sum of Squares SSTr
Error Sum of Squares SSE
14
Analysis of the Fixed Effects Model
  • Sum of Squares Partition for One Factor
    LayoutThis can be partitioned into two
    components SST SSTr SSE,where the sum of
    squares for treatments (SSTr)
  • measures the variability between the factor
    levels,
  • and the sum of squares for error (SSE)
  • measures the variability within the factor
    levels.

15
Analysis of the Fixed Effects Model
  • Sum of Squares Partition for One Factor Layout
  • On an intuitive level, the plausibility of the
    null hypothesis that the factor level means (µi)
    are all equal depends upon the relative size of
    the sum of squares for treatments (SSTr) to the
    sum of squares for error (SSE)

Possibly Likely the Same
Definitely NOT Likely the Same
Definitely VERY Likely the Same
16
Analysis of the Fixed Effects Model
  • F-Test for One Factor LayoutIn a one factor
    layout with k levels and n replications gives a
    total sample size kn N, the treatments are said
    to have k - 1 degrees of freedom and the error is
    said to have N - k degrees of freedom. Mean
    squares are obtained by dividing a sum of squares
    by its respective degrees of freedom so
    thatand

17
Analysis of the Fixed Effects Model
  • F-Test for One Factor LayoutA p-value for the
    null hypothesis that the factor level means µi,
    are all equal is calculated as
  • p-value P(X ³ F)
  • where the F-statistic isand the random
    variable X has a Fk-1, N - k distribution.

18
Analysis of the Fixed Effects Model
19
ANOVA Example
  • The tensile strength of a synthetic fiber used to
    make cloth for shirts is of interest to a
    manufacturer. It is suspected that strength is
    affected by the percentage of cotton in the
    fiber.
  • Five levels of cotton percentage are of interest
    15, 20, 25, 30, and 35.
  • Five observations are to be taken at each level
    of cotton percentage and the 25 total
    observations are to be run in random order.

20
ANOVA Example
RANDOMIZATIONPROCEDURE
. . .
. . .
. . .
21
ANOVA Example
Tensile Strength of Synthetic Fiber (lb/in2)
22
ANOVA Example
5-1 4
25-5 20
25-1 24
23
Critical Points for the F-Distribution
Alpha 0.05
24
ANOVA Example
25
ANOVA Example
26
Ishikawas Magnificent Seven Tools
  • The Seven Tools are
  • Histogram / Stem Leaf Diagram
  • Cause Effect (Fishbone) Diagram
  • Defect Concentration Diagram
  • Check Sheet
  • Scatter (Plot) Diagram
  • Pareto Chart
  • Control Chart - not covered on exam!
  • The tools were not invented by Ishikawa, but were
    very successfully put into methodical use by him
  • The first six are used before starting to use the
    seventh
  • They are also reused when needed to find an
    assignable cause

27
Ishikawas Tools Histogram
  • A histogram is a bar chart that takes the shape
    of the distribution of the data. The process for
    creating a histogram depends on the purpose for
    making the histogram.
  • One purpose of a histogram is to see the shape of
    a distribution. To do this, we would like to
    have as much data as possible, and use a fine
    resolution.
  • A second purpose of a histogram is to observe the
    frequency with which a class of problems occurs.
    The resolution is controlled by the number of
    problem classes.

28
Histogram of Lab 01 Results
29
Ishikawas Tools Fishbone Diagram
  • Cause Effect diagram constructed by
    brainstorming
  • Identified problem at the head
  • Connects potential causes along the spine
  • Sub-causes are listed along the major bones
  • Man
  • Material
  • Method
  • Machine
  • Environment

30
Cause Effect Diagram, Cont.
  • The purpose of the cause and effect diagram is to
    obtain as many potential influencers of a
    process, so that the problem solving can take a
    more directed approach.

31
Ishikawas Tools Defect Diagram
  • A defect concentration diagram graphically
    records the frequency of a defect with respect to
    product location.
  • Obtain a digital photo or multi-view part print
    showing all product faces.
  • Operator tallies the number and location of
    defects as they occur on the diagram.

32
Ishikawas Tools Check Sheet
  • Check sheets are used to collect data (values or
    pieces of information) in a consistent manner.
  • List each of the known / possible problems
  • Record each occurrence including time-orientation.

33
Ishikawas Tools Scatter Plot
  • A scatter plot shows the relationship between any
    two variables of interest
  • Plot one variable along the X-axis and the other
    along the Y-axis
  • The presence of a relationship can be inferred or
    ruled out, but it cannot determine if a cause and
    effect relationship exists

34
Ishikawas Tools Pareto Chart
  • 80 of any problem is the result of 20 of the
    potential causes
  • Histogram categories are sorted by the magnitude
    of the bar
  • A line graph is overlaid, and depicts the
    cumulative proportion of defects
  • Quickly identifies where to focus efforts

35
Use of Ishikawas Tools
  • Removing special causes of variation
  • Preparation for
  • hypothesis tests
  • control charts
  • process improvement

36
Questions Issues
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