ANOVA

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ANOVA

• HED 489 Biostatistics

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Randomized Design ANOVA

• Unlike Gosset, the inventor of analysis of
  varianceSir Ronald Fisherhas been inextricably
  linked to his invention. It is not by accident
  that the value calculated by ANOVA is called "F."
  In addition, again unlike Gosset, Fisher was a
  trained mathematician and worked as a
  statistician. About 1920, while working in
  agricultural research, Fisher developed analysis
  of variance. While Gosset was forced, by the
  nature of his brewery employer, to work with
  small samples, Fisher was under no such
  constraints. He was able to compare the effect of
  large numbers of various combinations of
  agricultural independent variables on his
  dependent variable, yield. All Fisher needed was
  farmland, of which England had plenty. What
  Fisher didn't have before he invented ANOVA, was
  a way to compare mean yield among three or more
  different plots of land.
• So, while there are many different types of
  analysis of variance, there are two things that
  set it apart from t-test you can analyze the
  effect of more than one independent variable, and
  you can study more than two samples. For example,
  with t-test, you could determine whether students
  learned more from Teacher A or Teacher B. But,
  you couldn't determine whether they learned more
  from Teacher A using the chalkboard or an
  overhead projector or Teacher B using the
  chalkboard or an overhead projector. These are
  the kinds of studies that ANOVA was invented to
  analyze.
• This first form of ANOVA is called "Randomized,"
  or "Completely Randomized." It assumes there is
  only one independent variable and, while it is
  usually shown being used with three or more
  groups, there is nothing to stop you from using
  it with two groups. Think of the Randomized ANOVA
  design as a large-sample t-test.

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Factorial Design ANOVA

• A factorial design analysis of variance allows
  you to compare the effect of more than one
  independent variable on your dependent variable
  or compare the effect of more than one level of
  independent variable on your dependent variable.
  If you look at the table, notice that we have two
  independent variablesthe teacher, and the method
  they use to teach. Note that each teacher will
  use each method. The six cells represent mean
  test scores for six different groups of students.
  Each group needs to have at least 20 students in
  order to use analysis of variance. It is
  important that you take particular note of how
  quickly the required number of students increases
  as you add independent variables or levels of
  independent variables. This has no statistical
  implications, but it sure makes it harder to
  conduct the study!

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• Analysis of variance, or "ANOVA" as it's commonly
  called, is a parametric test of differences you
  will use when
• You have more than two groups or samples of data
  
• You have one or two large groups of data, say
  more than 25 per group
• You have more than one independent variable
• Your independent variable has more than one
  level.
• You probably understand the first three
  conditions but perhaps not the fourth. For
  instance, you could use t-test to determine
  whether there is any difference in weight loss
  between people who exercise and those who don't.
  But, if you want to know whether there's any
  difference among those who exercise a little,
  those who exercise a lot, and those who don't
  exercise at all, you can't use t-test. See why
  not? Because you have three groups of data, and
  you can't use t-test with more than two groups.
  Enter ANOVA.

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Types of ANOVAs

• There are a LOT of different ways to use ANOVA.
  However, we're going to study only two
  typesCompletely Randomized Design, and
  Two-Factor Design. These are sufficiently
  complicated and once you've learned them, you'll
  have a good handle on ANOVA. It doesnt matter if
  your sample includes three, four, five, or more
  groups. The data are arrayed like an extended
  t-test 2independent samples.

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What ANOVA Does

• Look at the two little groups of data at the
  right. Consider how these data can vary
• Within each group, the values can vary from each
  other, and they do no two values in either group
  are the same
• Between each group, the mean of the groups can
  vary, and, again, in this example, they do
• And finally, totally, each value can vary from
  every other value in both groups.
• The question is are these variances sufficiently
  large such that we can assume the differences did
  not occur by chance alone? That is, are they
  statistically-significantly different. By
  calculating and comparing these three sources of
  variancetotal, between, and withinwe can make
  this determination.
• If you're confused, let me try to help you. Look
  at the little table below. Assume it represents
  the number of right answers from a 10-item quiz
  for three groups of students who were taught
  statistics by three different methods.

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• Now, do you see any variance in these scores?
  (The correct answer is, "No, I don't see any
  variance in these scores.") There isn't any. Is
  there any differences in how well the three
  groups did? Of course not, they each did equally
  wellperfectly well, in fact. You don't have to
  be a rocket scientist to be able to figure out
  that there's no statistical difference among the
  three groups, right? You accept the null
  hypothesisno difference in teaching method.
• The chances of this happening are, obviously,
  very small. And, the greater the differencesthat
  is, variancethe greater the chances of
  statistical significance. But, how much variance
  is needed to reach statistical significance?
  That's where ANOVA comes in.

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Degrees of Freedom

• Before we get more information about ANOVA we
  need to speak about Degrees of Freedom. Degrees
  of Freedoms (DF) is important to help determine
  whether there is a difference between groups.

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Calculating Degrees of Freedom
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The Formula

• In just a second, you're going to see the ANOVA
  formula. It's pretty big, but it's not that bad.
  However, be sure you write down the notation or
  you'll be lost until you get familiar with it. An
  example or two
• If you see the summation sign followed by j, you
  would sum the scores in each group.
• If you see the summation sign followed by ij, you
  would sum all the scores, across all groups.

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• In just a second, you're going to see the ANOVA
  formula. It's pretty big, but it's not that bad.
  However, be sure you write down the notation or
  you'll be lost until you get familiar with it. An
  example or two
• If you see the summation sign followed by j, you
  would sum the scores in each group.
• If you see the summation sign followed by ij, you
  would sum all the scores, across all groups.

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The Formula
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• I know the formula looks a little spooky, but
  look at the formula for standard deviation. Now,
  look at the ANOVA formula. Isn't it true that the
  numerator and the two major areas in the
  denominator are very similar to the standard
  deviation formula. What we're really doing in
  calculating three variances, subtracting one from
  the other in the denominator, and dividing it
  into the numerator.
• There are three steps in calculating ANOVA, and
  these are shown to the left of the formula
• Because we do so much squaring and summing of
  values, the first step is to calculate "Sum of
  Squares" for the total, between groups, and
  within groups variance.
• We then calculate the average or "Mean Square"
  variances by dividing each Sum of Squares by its
  respective degrees of freedom essentially, by
  the number of values associated with each source
  of variance.
• Finally, we divide the Mean Square Between Groups
  variance by the Mean Square Within Groups to
  calculate F. Because of the different values you
  calculate, you construct a "Summary Table" that
  show Sum of Squares, Mean Squares, degrees of
  freedom, F or F's if more than one, and the
  probability level or levels. From there, it's a
  simple matter to go to the F table and check for
  significance.
• I'm going to treat ANOVA just like I did t-test.
  I'm going to take you through each step,
  explaining what's being calculated along the way.
  I'll use the example in the book and repeat the
  values calculated at each step in the middle
  frame at the right. BUT REMEMBERthis can be
  done using Excel!

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Step 1

• The example the book uses sets out to determine
  if different levels of shocking movies result in
  any difference in the amount of mistakes students
  will make during a driving simulation.
• The null hypothesis for this study is
• H0 There is no difference in the amount of
  mistakes made during the driving simulation among
  different levels of shocking movies.
• The alternative hypothesis is
• H1 There is a difference in the amount of
  mistakes made during the driving simulation among
  different levels of shocking movies.

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This particular driver education teacher wants to
see if watching a video with various levels of
shock will reduce mistakes on the driving
stimulator. One film has no shock valuethe
other three have varying levels of shock, with
the high shock showing graphic blood, accidents,
and dismembered body parts due to poor driving.
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• Click here to go through an excel demonstration
  on how to determine the F in this case. Note the
  bottom has folders with the Steps. Start with
  Step 1, end with Step 13

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• Go find the value associated with 3 and 44
  degrees of freedom. Note that this table doesn't
  have the value for 44 degrees of freedom. When
  this happens, use the next smaller degrees of
  freedom. This will result in larger critical
  values, and will minimize your risk of Type 1
  error.
• When you get to the cell that intersects df1 and
  df2, look at the value associated with the
  probability that you've selected. Usually, this
  will be .05. If the value you calculated is
  greater than the tabled value, you have found a
  statistically-significant difference.
• The reason tabled degrees of freedom are called 1
  and 2, rather than b and w, is because other,
  more-complicated forms of ANOVA actually produce
  more than one F-ratio and, therefore, different
  pairs of degrees of freedom that have to be
  applied to the F-distribution table.
• Finally, notice that although the example in
  Kuzma probably used a probability level of .05,
  you report the greatest level of probability that
  results. The calculated F-ratio in this example
  is 14.71, and the largest critical value under df
  3 and 44 is 6.60 at p.001. So, while the
  calculated F is larger than the critical value at
  p.05, it's also larger than the value at p.001.
  Therefore, you report that your calculated ratio
  is significant at plt.001.
• Before we forget, you would reject the null
  hypothesis, accept the alternative hypothesis,
  and conclude that different shock levels do,
  indeed, have an impact on problem solving.

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• OK, now what idiot would do this by hand when we
  have computers. So, lets go through how we can
  do this via Excel.
• Click here to go to that tutorial (if you are
  using Office 2007, click here).
• After reviewing the above tutorial, go to next
  slide for assignment.

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• Click here to go to an excel file with data.
  Calculate the results.