Presentation on theme: "Overview of Lecture Partitioning Evaluating the Null Hypothesis ANOVA"— Presentation transcript:
1 Overview of LecturePartitioningEvaluating the Null HypothesisANOVABasic RatiosSums of SquaresMean SquaresDegrees of FreedomF-RatioEvaluating the F-RatioAnalytical Comparisons
2 Partitioning the Deviations AS25A2T-Within SubjectsdeviationBetween Subjects
3 Evaluating the Null Hypothesis If we consider the ratio of the between groups variability and the within groups variabilityThen we have
4 Evaluating the Null Hypothesis If the null hypothesis is true then the treatment effect is equal to zero:If the null hypothesis is false then the treatment effect is greater than zero:
5 ANOVAAnalysis of variance uses the ratio of two sources of variability to test the null hypothesisBetween group variability estimates both experimental error and treatment effectsWithin subjects variability estimates experimental error
6 From Deviations to Variances In order to evaluate the null hypothesis, it is necessary to transform the between group and within-group deviations into more useful quantities, namely, variances.For this reason the statistical analysis involving the comparison of variances reflecting different sources of variability is called the analysis of variance.For the purposes of analysis of variance a variance is defined as follows
7 The Sums of Squares From the basic deviations A similar relationship holds for the sum of squaresIn other words
9 Basic Ratios:A Basic Ratio is defined as:The numerator term for any basic ratio involves two steps:the initial squaring of a set of quantitiessumming the squared quantities if more than one is present.The denominator term for each ratio is the number of items that contribute to the sum or score.Basic Ratios make the calculation of the sum of squares relatively simple
10 The Notation for Basic Ratios To emphasise the critical nature of these basic ratios, we will use a distinctive symbol to designate basic ratios and to distinguish among them
11 Calculating the Basic Ratios Starting with the basic score, AS, and substituting into the formula for basic ratios:The second basic ratio involves the sums of the treatment conditions.The final basic ratio we require involves the grand sum.
12 Calculating the Sums of Squares The sums of squares can be calculated by combining these basic ratios.Total Sum of SquaresBetween Group Sum of SquaresWithin Group Sum of Squares
13 Variance Estimates: Mean Squares The ratio we are interested in is the ratio of the between groups variability and the within groups variabilityIn this context, the variability is defined by the equation.where SS refers to the component sums of squares and df represent the degrees of freedom associated with the SS.
14 Degrees of FreedomThe degrees of freedom associated with a sum of squares correspond to the number of scores with independent information which enter into the calculation of the sum of squares.Degrees of freedom are the number of observations that are free to vary when we know something about those observations.
15 Degrees of FreedomThe Between Group Degrees of FreedomThe Within Group Degrees of FreedomThe Total Degrees of Freedom
16 Mean SquaresThe Between Group Mean SquareThe Within Group Mean Square
18 The Anova Summary Table The results of the Anova are usually displayed by computers programs in a summary table:
19 Testing the Null Hypothesis In order to decide whether or not the null hypothesis is rejected we need to find out what value of F is necessary to reject the null hypothesis.There is a simple rule for this.Reject H0 when Fobserved> Fcritical otherwise do not reject H0We obtain a value for Fcritical by looking it up in the F tables.
20 The Critical F-ValueTo find the critical valueTake the degrees of freedom for the effect (A) and look along the horizontal axis of the F table.Take the degrees of freedom for the error term (S/A) and look down the vertical axis of the F table.The place were the column for the degrees of freedom of the effect A meets the row for the degrees of freedom of the error (S/A) is the critical value of F.For these data Fcritical =3.89 so we can reject the null hypothesis
21 The Omnibus FThe F-ratio includes information about all the levels of the independent variable that we have manipulated.The F-ratio for an overall difference between the means as reported in the ANOVA summary table is known as the Omnibus F-ratio.The best the Omnibus F-ratio can tell us is that there are differences between the means.It cannot tell us that what those differences are.
22 Analytical Comparisons With a nonsignificant omnibus F we are prepared to assert that there are no observed differences among the means.We can stop the analysis there.A significant omnibus F demands further analysis of the data.Which differences between the means are real and which are not?
23 Analytical Comparisons There are two basic approaches to solving the problem.Before we set out to collect the data, we could have made specific predictions about the direction of the effectsIn this case we can use a technique known as planned (a priori) comparisons.We might have designed an experiment where we couldn't be precise enough to say what the differences would be.In this case we use post hoc (after the event) comparisons.
24 Planned (A Priori) Comparisons For example, assume the three levels of the independent variable are lecturer style:A1: Lectures with worksheets.A2: Lectures only.A3: No lectures, only worksheets.From previous research, we anticipate that A1 > A2, A1 > A3 (we are making no predictions about A2 vs A3).Does the data support this?
25 Differences as the sum of weighted means Let us adopt the symbol y to represent the difference we are interested in:Y = mA1 – mA2We can rewrite this as:Y = (1) mA1 +(–1) mA2Including all the means in the experiment:Y = (1) mA1 +(–1) mA2 +(0) mA3
26 Sum of Squares for a planned comparison Planned comparisons are based on the calculation of an F-ratio.This requires us to calculate the variability inherent in the comparisonWe calculate a sum of squares associated with the comparison. This is given by:
27 Sum of Squares of a planned comparison The sum of squares of the comparison is given by
28 Testing the planned comparison A F ratio is calculated to test the comparison. For this a mean square is required.The mean square for the comparison is calculated by:All planned comparisons have on 1 degree of freedom.The F-ratio is calculated by:
29 Evaluating the planned comparison’s null hypothesis Critical F's for comparisons use the degrees of freedom for the numerator and the denominator of the F-ratio.There are 1 and 12 degrees of freedom for this comparison.Fcritical(1, 12) for p≤0.05=4.75Given that Fobservedl=14.29, we can reject the null hypothesis and conclude that A1 (lectures with worksheets) leads to better scores than A2 (lectures only).
30 Next WeekPost hoc comparisonsTesting the assumptions that underlie ANOVATwo computer programs for analysing a one-way between groups analysis of variance