The Situation Test Statistic Computing the Quantities ANOVA Terminology 1. I populations or treatments of equal size J are to be compared. 2. µ i denotes the actual mean of the i th population. 3. Null hypothesis. H 0 : µ 1 = µ 2 = ··· = µ I (no difference, or, no effect) 4. Alternative hypothesis. H a : At least two means differ. 5. For example, if among 10 pain relievers, all have a sample average time until pain lessens of around 20 minutes and one has a sample average of around 10 minutes, then it pretty much looks like that one is different. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities ANOVA Terminology 1. I populations or treatments of equal size J are to be compared. 2. µ i denotes the actual mean of the i th population. 3. Null hypothesis. H 0 : µ 1 = µ 2 = ··· = µ I (no difference, or, no effect) 4. Alternative hypothesis. H a : At least two means differ. 5. For example, if among 10 pain relievers, all have a sample average time until pain lessens of around 20 minutes and one has a sample average of around 10 minutes, then it pretty much looks like that one is different. When it’s not that obvious, we need a testing procedure (finer analysis). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities ANOVA Terminology (cont.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities ANOVA Terminology (cont.) 6. X i , j is the random variable that denotes the j th measurement from the i th population/treatment group. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities ANOVA Terminology (cont.) 6. X i , j is the random variable that denotes the j th measurement from the i th population/treatment group. x i , j will be the observed value (“as always”) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities ANOVA Terminology (cont.) 6. X i , j is the random variable that denotes the j th measurement from the i th population/treatment group. x i , j will be the observed value (“as always”) Data is often displayed in a matrix. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities ANOVA Terminology (cont.) 6. X i , j is the random variable that denotes the j th measurement from the i th population/treatment group. x i , j will be the observed value (“as always”) Data is often displayed in a matrix. 7. Individual sample means: X i · = ∑ J j = 1 X ij J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities ANOVA Terminology (cont.) 6. X i , j is the random variable that denotes the j th measurement from the i th population/treatment group. x i , j will be the observed value (“as always”) Data is often displayed in a matrix. 7. Individual sample means: X i · = ∑ J j = 1 X ij J The dot says we summed over the second variable. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities ANOVA Terminology (cont.) 6. X i , j is the random variable that denotes the j th measurement from the i th population/treatment group. x i , j will be the observed value (“as always”) Data is often displayed in a matrix. 7. Individual sample means: X i · = ∑ J j = 1 X ij J The dot says we summed over the second variable. � 2 i = ∑ J � X ij − X i · j = 1 8. Sample variance: S 2 J − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities ANOVA Terminology (cont.) 6. X i , j is the random variable that denotes the j th measurement from the i th population/treatment group. x i , j will be the observed value (“as always”) Data is often displayed in a matrix. 7. Individual sample means: X i · = ∑ J j = 1 X ij J The dot says we summed over the second variable. � 2 i = ∑ J � X ij − X i · j = 1 8. Sample variance: S 2 J − 1 9. Grand mean: X ·· = ∑ I i = 1 ∑ J j = 1 X ij IJ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Underlying Assumptions and Their Consequences logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Underlying Assumptions and Their Consequences 1. All populations are assumed to be normally distributed with the same variance σ 2 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Underlying Assumptions and Their Consequences 1. All populations are assumed to be normally distributed with the same variance σ 2 . Hence all X ij are normally distributed and E ( X ij ) = µ i and V ( X ij ) = σ 2 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Underlying Assumptions and Their Consequences 1. All populations are assumed to be normally distributed with the same variance σ 2 . Hence all X ij are normally distributed and E ( X ij ) = µ i and V ( X ij ) = σ 2 . 2. If the largest sample standard deviation is at most twice the smallest sample standard deviation logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Underlying Assumptions and Their Consequences 1. All populations are assumed to be normally distributed with the same variance σ 2 . Hence all X ij are normally distributed and E ( X ij ) = µ i and V ( X ij ) = σ 2 . 2. If the largest sample standard deviation is at most twice the smallest sample standard deviation, then it is (still) reasonable to assume that the σ s are equal. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Underlying Assumptions and Their Consequences 1. All populations are assumed to be normally distributed with the same variance σ 2 . Hence all X ij are normally distributed and E ( X ij ) = µ i and V ( X ij ) = σ 2 . 2. If the largest sample standard deviation is at most twice the smallest sample standard deviation, then it is (still) reasonable to assume that the σ s are equal. 3. To check normality, use a normal probability plot. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Underlying Assumptions and Their Consequences 1. All populations are assumed to be normally distributed with the same variance σ 2 . Hence all X ij are normally distributed and E ( X ij ) = µ i and V ( X ij ) = σ 2 . 2. If the largest sample standard deviation is at most twice the smallest sample standard deviation, then it is (still) reasonable to assume that the σ s are equal. 3. To check normality, use a normal probability plot. 4. If the null hypothesis µ 1 = µ 2 = ··· = µ I is true logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Underlying Assumptions and Their Consequences 1. All populations are assumed to be normally distributed with the same variance σ 2 . Hence all X ij are normally distributed and E ( X ij ) = µ i and V ( X ij ) = σ 2 . 2. If the largest sample standard deviation is at most twice the smallest sample standard deviation, then it is (still) reasonable to assume that the σ s are equal. 3. To check normality, use a normal probability plot. 4. If the null hypothesis µ 1 = µ 2 = ··· = µ I is true, then all sample averages should be close to each other. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Underlying Assumptions and Their Consequences 1. All populations are assumed to be normally distributed with the same variance σ 2 . Hence all X ij are normally distributed and E ( X ij ) = µ i and V ( X ij ) = σ 2 . 2. If the largest sample standard deviation is at most twice the smallest sample standard deviation, then it is (still) reasonable to assume that the σ s are equal. 3. To check normality, use a normal probability plot. 4. If the null hypothesis µ 1 = µ 2 = ··· = µ I is true, then all sample averages should be close to each other. 5. To determine if the variation is consistent with the null hypothesis logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Underlying Assumptions and Their Consequences 1. All populations are assumed to be normally distributed with the same variance σ 2 . Hence all X ij are normally distributed and E ( X ij ) = µ i and V ( X ij ) = σ 2 . 2. If the largest sample standard deviation is at most twice the smallest sample standard deviation, then it is (still) reasonable to assume that the σ s are equal. 3. To check normality, use a normal probability plot. 4. If the null hypothesis µ 1 = µ 2 = ··· = µ I is true, then all sample averages should be close to each other. 5. To determine if the variation is consistent with the null hypothesis, we compare a measure of the variance between the samples logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Underlying Assumptions and Their Consequences 1. All populations are assumed to be normally distributed with the same variance σ 2 . Hence all X ij are normally distributed and E ( X ij ) = µ i and V ( X ij ) = σ 2 . 2. If the largest sample standard deviation is at most twice the smallest sample standard deviation, then it is (still) reasonable to assume that the σ s are equal. 3. To check normality, use a normal probability plot. 4. If the null hypothesis µ 1 = µ 2 = ··· = µ I is true, then all sample averages should be close to each other. 5. To determine if the variation is consistent with the null hypothesis, we compare a measure of the variance between the samples (“between-samples” variation) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Underlying Assumptions and Their Consequences 1. All populations are assumed to be normally distributed with the same variance σ 2 . Hence all X ij are normally distributed and E ( X ij ) = µ i and V ( X ij ) = σ 2 . 2. If the largest sample standard deviation is at most twice the smallest sample standard deviation, then it is (still) reasonable to assume that the σ s are equal. 3. To check normality, use a normal probability plot. 4. If the null hypothesis µ 1 = µ 2 = ··· = µ I is true, then all sample averages should be close to each other. 5. To determine if the variation is consistent with the null hypothesis, we compare a measure of the variance between the samples (“between-samples” variation) to a measure of the variation “within” the samples. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Underlying Assumptions and Their Consequences 1. All populations are assumed to be normally distributed with the same variance σ 2 . Hence all X ij are normally distributed and E ( X ij ) = µ i and V ( X ij ) = σ 2 . 2. If the largest sample standard deviation is at most twice the smallest sample standard deviation, then it is (still) reasonable to assume that the σ s are equal. 3. To check normality, use a normal probability plot. 4. If the null hypothesis µ 1 = µ 2 = ··· = µ I is true, then all sample averages should be close to each other. 5. To determine if the variation is consistent with the null hypothesis, we compare a measure of the variance between the samples (“between-samples” variation) to a measure of the variation “within” the samples. (Remember that we assume all populations have the same σ ). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities 1. Mean square for treatments. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities 1. Mean square for treatments. J � 2 + ··· + �� � 2 � � MSTr = X 1 · − X ·· X I · − X ·· I − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities 1. Mean square for treatments. I J � 2 + ··· + J �� � 2 � � 2 � ∑ � MSTr = X 1 · − X ·· X I · − X ·· = X i · − X ·· I − 1 I − 1 i = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities 1. Mean square for treatments. I J � 2 + ··· + J �� � 2 � � 2 � ∑ � MSTr = X 1 · − X ·· X I · − X ·· = X i · − X ·· I − 1 I − 1 i = 1 2. Mean square for error logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities 1. Mean square for treatments. I J � 2 + ··· + J �� � 2 � � 2 � ∑ � MSTr = X 1 · − X ·· X I · − X ·· = X i · − X ·· I − 1 I − 1 i = 1 2. Mean square for error MSE = S 2 1 + ··· + S 2 I . I logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities 1. Mean square for treatments. I J � 2 + ··· + J �� � 2 � � 2 � ∑ � MSTr = X 1 · − X ·· X I · − X ·· = X i · − X ·· I − 1 I − 1 i = 1 2. Mean square for error MSE = S 2 1 + ··· + S 2 I . I 3. The test statistic for single factor ANOVA is F = MSTr MSE . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities 1. Mean square for treatments. I J � 2 + ··· + J �� � 2 � � 2 � ∑ � MSTr = X 1 · − X ·· X I · − X ·· = X i · − X ·· I − 1 I − 1 i = 1 2. Mean square for error MSE = S 2 1 + ··· + S 2 I . I 3. The test statistic for single factor ANOVA is F = MSTr MSE . 4. The J in MSTr re-scales the spread of the means back to the spread of individual samples. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities 1. Mean square for treatments. I J � 2 + ··· + J �� � 2 � � 2 � ∑ � MSTr = X 1 · − X ·· X I · − X ·· = X i · − X ·· I − 1 I − 1 i = 1 2. Mean square for error MSE = S 2 1 + ··· + S 2 I . I 3. The test statistic for single factor ANOVA is F = MSTr MSE . 4. The J in MSTr re-scales the spread of the means back to the spread of individual samples. 5. If the null hypothesis is true: E ( MSTr ) = E ( MSE ) = σ 2 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities 1. Mean square for treatments. I J � 2 + ··· + J �� � 2 � � 2 � ∑ � MSTr = X 1 · − X ·· X I · − X ·· = X i · − X ·· I − 1 I − 1 i = 1 2. Mean square for error MSE = S 2 1 + ··· + S 2 I . I 3. The test statistic for single factor ANOVA is F = MSTr MSE . 4. The J in MSTr re-scales the spread of the means back to the spread of individual samples. 5. If the null hypothesis is true: E ( MSTr ) = E ( MSE ) = σ 2 . If the null hypothesis is false: E ( MSTr ) > E ( MSE ) = σ 2 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities 1. Mean square for treatments. I J � 2 + ··· + J �� � 2 � � 2 � ∑ � MSTr = X 1 · − X ·· X I · − X ·· = X i · − X ·· I − 1 I − 1 i = 1 2. Mean square for error MSE = S 2 1 + ··· + S 2 I . I 3. The test statistic for single factor ANOVA is F = MSTr MSE . 4. The J in MSTr re-scales the spread of the means back to the spread of individual samples. 5. If the null hypothesis is true: E ( MSTr ) = E ( MSE ) = σ 2 . If the null hypothesis is false: E ( MSTr ) > E ( MSE ) = σ 2 . 6. When the null hypothesis is true, the statistic F = MSTr MSE has an F -distribution with ν 1 = I − 1 and ν 2 = I ( J − 1 ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities 1. Mean square for treatments. I J � 2 + ··· + J �� � 2 � � 2 � ∑ � MSTr = X 1 · − X ·· X I · − X ·· = X i · − X ·· I − 1 I − 1 i = 1 2. Mean square for error MSE = S 2 1 + ··· + S 2 I . I 3. The test statistic for single factor ANOVA is F = MSTr MSE . 4. The J in MSTr re-scales the spread of the means back to the spread of individual samples. 5. If the null hypothesis is true: E ( MSTr ) = E ( MSE ) = σ 2 . If the null hypothesis is false: E ( MSTr ) > E ( MSE ) = σ 2 . 6. When the null hypothesis is true, the statistic F = MSTr MSE has an F -distribution with ν 1 = I − 1 and ν 2 = I ( J − 1 ) . 7. A rejection region f > F α , I − 1 , I ( J − 1 ) gives a test of significance level α . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities 1. Mean square for treatments. I J � 2 + ··· + J �� � 2 � � 2 � ∑ � MSTr = X 1 · − X ·· X I · − X ·· = X i · − X ·· I − 1 I − 1 i = 1 2. Mean square for error MSE = S 2 1 + ··· + S 2 I . I 3. The test statistic for single factor ANOVA is F = MSTr MSE . 4. The J in MSTr re-scales the spread of the means back to the spread of individual samples. 5. If the null hypothesis is true: E ( MSTr ) = E ( MSE ) = σ 2 . If the null hypothesis is false: E ( MSTr ) > E ( MSE ) = σ 2 . 6. When the null hypothesis is true, the statistic F = MSTr MSE has an F -distribution with ν 1 = I − 1 and ν 2 = I ( J − 1 ) . 7. A rejection region f > F α , I − 1 , I ( J − 1 ) gives a test of significance level α . 8. For p -values, use the area to the right of the test statistic. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Example. Perform an ANOVA on the enclosed test data to see if the “true average performances” can be considered equal. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Example. Perform an ANOVA on the enclosed test data to see if the “true average performances” can be considered equal. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Keeping Track of the Data logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Keeping Track of the Data The key to ANOVA (by hand) is orderly bookkeeping. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Keeping Track of the Data The key to ANOVA (by hand) is orderly bookkeeping. Also remember that all this was done before computers. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Keeping Track of the Data The key to ANOVA (by hand) is orderly bookkeeping. Also remember that all this was done before computers. So anything that could save a few operations logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Keeping Track of the Data The key to ANOVA (by hand) is orderly bookkeeping. Also remember that all this was done before computers. So anything that could save a few operations, or help minimize rounding errors logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Keeping Track of the Data The key to ANOVA (by hand) is orderly bookkeeping. Also remember that all this was done before computers. So anything that could save a few operations, or help minimize rounding errors, was appreciated. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities I J ∑ ∑ 1. Grand total: x ·· = x ij i = 1 j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities I J ∑ ∑ 1. Grand total: x ·· = x ij i = 1 j = 1 I J I J ij − 1 ( x ij − x ·· ) 2 = x 2 IJ x 2 ∑ ∑ ∑ ∑ 2. Total sum of squares: SST = ·· i = 1 j = 1 i = 1 j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities I J ∑ ∑ 1. Grand total: x ·· = x ij i = 1 j = 1 I J I J ij − 1 ( x ij − x ·· ) 2 = x 2 IJ x 2 ∑ ∑ ∑ ∑ 2. Total sum of squares: SST = ·· i = 1 j = 1 i = 1 j = 1 I J I ( x i · − x ·· ) 2 = 1 i · − 1 ∑ ∑ ∑ x 2 IJ x 2 3. Treatment sum of squares: SSTr = ·· , J i = 1 j = 1 i = 1 J ∑ where x i · = x ij j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities I J ∑ ∑ 1. Grand total: x ·· = x ij i = 1 j = 1 I J I J ij − 1 ( x ij − x ·· ) 2 = x 2 IJ x 2 ∑ ∑ ∑ ∑ 2. Total sum of squares: SST = ·· i = 1 j = 1 i = 1 j = 1 I J I ( x i · − x ·· ) 2 = 1 i · − 1 ∑ ∑ ∑ x 2 IJ x 2 3. Treatment sum of squares: SSTr = ·· , J i = 1 j = 1 i = 1 J ∑ where x i · = x ij j = 1 I J ( x ij − x i · ) 2 ∑ ∑ 4. Error sum of squares: SSE = i = 1 j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities I J ∑ ∑ 1. Grand total: x ·· = x ij i = 1 j = 1 I J I J ij − 1 ( x ij − x ·· ) 2 = x 2 IJ x 2 ∑ ∑ ∑ ∑ 2. Total sum of squares: SST = ·· i = 1 j = 1 i = 1 j = 1 I J I ( x i · − x ·· ) 2 = 1 i · − 1 ∑ ∑ ∑ x 2 IJ x 2 3. Treatment sum of squares: SSTr = ·· , J i = 1 j = 1 i = 1 J ∑ where x i · = x ij j = 1 I J ( x ij − x i · ) 2 ∑ ∑ 4. Error sum of squares: SSE = i = 1 j = 1 5. MSTr = SSTr I − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities I J ∑ ∑ 1. Grand total: x ·· = x ij i = 1 j = 1 I J I J ij − 1 ( x ij − x ·· ) 2 = x 2 IJ x 2 ∑ ∑ ∑ ∑ 2. Total sum of squares: SST = ·· i = 1 j = 1 i = 1 j = 1 I J I ( x i · − x ·· ) 2 = 1 i · − 1 ∑ ∑ ∑ x 2 IJ x 2 3. Treatment sum of squares: SSTr = ·· , J i = 1 j = 1 i = 1 J ∑ where x i · = x ij j = 1 I J ( x ij − x i · ) 2 ∑ ∑ 4. Error sum of squares: SSE = i = 1 j = 1 5. MSTr = SSTr I − 1 (What happened to J ? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities I J ∑ ∑ 1. Grand total: x ·· = x ij i = 1 j = 1 I J I J ij − 1 ( x ij − x ·· ) 2 = x 2 IJ x 2 ∑ ∑ ∑ ∑ 2. Total sum of squares: SST = ·· i = 1 j = 1 i = 1 j = 1 I J I ( x i · − x ·· ) 2 = 1 i · − 1 ∑ ∑ ∑ x 2 IJ x 2 3. Treatment sum of squares: SSTr = ·· , J i = 1 j = 1 i = 1 J ∑ where x i · = x ij j = 1 I J ( x ij − x i · ) 2 ∑ ∑ 4. Error sum of squares: SSE = i = 1 j = 1 5. MSTr = SSTr I − 1 (What happened to J ? It’s the dummy sum over j !) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities I J ∑ ∑ 1. Grand total: x ·· = x ij i = 1 j = 1 I J I J ij − 1 ( x ij − x ·· ) 2 = x 2 IJ x 2 ∑ ∑ ∑ ∑ 2. Total sum of squares: SST = ·· i = 1 j = 1 i = 1 j = 1 I J I ( x i · − x ·· ) 2 = 1 i · − 1 ∑ ∑ ∑ x 2 IJ x 2 3. Treatment sum of squares: SSTr = ·· , J i = 1 j = 1 i = 1 J ∑ where x i · = x ij j = 1 I J ( x ij − x i · ) 2 ∑ ∑ 4. Error sum of squares: SSE = i = 1 j = 1 5. MSTr = SSTr I − 1 (What happened to J ? It’s the dummy sum over j !) SSE 6. MSE = I ( J − 1 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities I J ∑ ∑ 1. Grand total: x ·· = x ij i = 1 j = 1 I J I J ij − 1 ( x ij − x ·· ) 2 = x 2 IJ x 2 ∑ ∑ ∑ ∑ 2. Total sum of squares: SST = ·· i = 1 j = 1 i = 1 j = 1 I J I ( x i · − x ·· ) 2 = 1 i · − 1 ∑ ∑ ∑ x 2 IJ x 2 3. Treatment sum of squares: SSTr = ·· , J i = 1 j = 1 i = 1 J ∑ where x i · = x ij j = 1 I J ( x ij − x i · ) 2 ∑ ∑ 4. Error sum of squares: SSE = i = 1 j = 1 5. MSTr = SSTr I − 1 (What happened to J ? It’s the dummy sum over j !) SSE F = MSTr 6. MSE = I ( J − 1 ) , MSE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Are the Claimed Formulas Right? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Are the Claimed Formulas Right? SST logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Are the Claimed Formulas Right? I J ( x ij − x ·· ) 2 ∑ ∑ = SST i = 1 j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Are the Claimed Formulas Right? I J I J ( x ij − x ·· ) 2 = x 2 ij − 2 x ij x ·· + x 2 ∑ ∑ ∑ ∑ = SST ·· i = 1 j = 1 i = 1 j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Are the Claimed Formulas Right? I J I J ( x ij − x ·· ) 2 = x 2 ij − 2 x ij x ·· + x 2 ∑ ∑ ∑ ∑ = SST ·· i = 1 j = 1 i = 1 j = 1 I J I J I J ∑ ∑ x 2 ∑ ∑ ∑ ∑ x 2 = ij − 2 x ·· x ij + ·· i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Are the Claimed Formulas Right? I J I J ( x ij − x ·· ) 2 = x 2 ij − 2 x ij x ·· + x 2 ∑ ∑ ∑ ∑ = SST ·· i = 1 j = 1 i = 1 j = 1 I J I J I J ∑ ∑ x 2 ∑ ∑ ∑ ∑ x 2 = ij − 2 x ·· x ij + ·· i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 � 2 � I J I J I J I J ij − 2 1 ∑ ∑ x 2 ∑ ∑ ∑ ∑ ∑ ∑ = x ij + IJ x ij x ij IJ IJ i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Are the Claimed Formulas Right? I J I J ( x ij − x ·· ) 2 = x 2 ij − 2 x ij x ·· + x 2 ∑ ∑ ∑ ∑ = SST ·· i = 1 j = 1 i = 1 j = 1 I J I J I J ∑ ∑ x 2 ∑ ∑ ∑ ∑ x 2 = ij − 2 x ·· x ij + ·· i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 � 2 � I J I J I J I J ij − 2 1 ∑ ∑ x 2 ∑ ∑ ∑ ∑ ∑ ∑ = x ij + IJ x ij x ij IJ IJ i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 I J I J I J ij − 1 x 2 ∑ ∑ ∑ ∑ ∑ ∑ = x ij x ij IJ i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Are the Claimed Formulas Right? I J I J ( x ij − x ·· ) 2 = x 2 ij − 2 x ij x ·· + x 2 ∑ ∑ ∑ ∑ = SST ·· i = 1 j = 1 i = 1 j = 1 I J I J I J ∑ ∑ x 2 ∑ ∑ ∑ ∑ x 2 = ij − 2 x ·· x ij + ·· i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 � 2 � I J I J I J I J ij − 2 1 ∑ ∑ x 2 ∑ ∑ ∑ ∑ ∑ ∑ = x ij + IJ x ij x ij IJ IJ i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 I J I J I J I J ij − 1 ij − 1 x 2 x 2 IJx 2 ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ = x ij = x ij IJ ·· i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Are the Claimed Formulas Right? I J I J ( x ij − x ·· ) 2 = x 2 ij − 2 x ij x ·· + x 2 ∑ ∑ ∑ ∑ = SST ·· i = 1 j = 1 i = 1 j = 1 I J I J I J ∑ ∑ x 2 ∑ ∑ ∑ ∑ x 2 = ij − 2 x ·· x ij + ·· i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 � 2 � I J I J I J I J ij − 2 1 ∑ ∑ x 2 ∑ ∑ ∑ ∑ ∑ ∑ = x ij + IJ x ij x ij IJ IJ i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 I J I J I J I J ij − 1 ij − 1 x 2 x 2 IJx 2 ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ = x ij = x ij IJ ·· i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 Treatment sum of squares: Similar. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Fundamental Identity: SST = SSTr + SSE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Fundamental Identity: SST = SSTr + SSE x ij − x ·· logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Fundamental Identity: SST = SSTr + SSE = ( x ij − x i · )+( x i · − x ·· ) x ij − x ·· logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Fundamental Identity: SST = SSTr + SSE = ( x ij − x i · )+( x i · − x ·· ) x ij − x ·· ( x ij − x ·· ) 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Fundamental Identity: SST = SSTr + SSE = ( x ij − x i · )+( x i · − x ·· ) x ij − x ·· ( x ij − x i · ) 2 + 2 ( x ij − x i · )( x i · − x ·· )+( x i · − x ·· ) 2 ( x ij − x ·· ) 2 = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Fundamental Identity: SST = SSTr + SSE = ( x ij − x i · )+( x i · − x ·· ) x ij − x ·· ( x ij − x i · ) 2 + 2 ( x ij − x i · )( x i · − x ·· )+( x i · − x ·· ) 2 ( x ij − x ·· ) 2 = Now sum over i,j. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Fundamental Identity: SST = SSTr + SSE = ( x ij − x i · )+( x i · − x ·· ) x ij − x ·· ( x ij − x i · ) 2 + 2 ( x ij − x i · )( x i · − x ·· )+( x i · − x ·· ) 2 ( x ij − x ·· ) 2 = Now sum over i,j. The middle term drops out after summing over j logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Fundamental Identity: SST = SSTr + SSE = ( x ij − x i · )+( x i · − x ·· ) x ij − x ·· ( x ij − x i · ) 2 + 2 ( x ij − x i · )( x i · − x ·· )+( x i · − x ·· ) 2 ( x ij − x ·· ) 2 = Now sum over i,j. The middle term drops out after summing J ∑ over j , because ( x ij − x i · ) = 0. j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Fundamental Identity: SST = SSTr + SSE = ( x ij − x i · )+( x i · − x ·· ) x ij − x ·· ( x ij − x i · ) 2 + 2 ( x ij − x i · )( x i · − x ·· )+( x i · − x ·· ) 2 ( x ij − x ·· ) 2 = Now sum over i,j. The middle term drops out after summing J ∑ over j , because ( x ij − x i · ) = 0. Hence j = 1 SST logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Fundamental Identity: SST = SSTr + SSE = ( x ij − x i · )+( x i · − x ·· ) x ij − x ·· ( x ij − x i · ) 2 + 2 ( x ij − x i · )( x i · − x ·· )+( x i · − x ·· ) 2 ( x ij − x ·· ) 2 = Now sum over i,j. The middle term drops out after summing J ∑ over j , because ( x ij − x i · ) = 0. Hence j = 1 I J ( x ij − x ·· ) 2 ∑ ∑ SST = i = 1 j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Fundamental Identity: SST = SSTr + SSE = ( x ij − x i · )+( x i · − x ·· ) x ij − x ·· ( x ij − x i · ) 2 + 2 ( x ij − x i · )( x i · − x ·· )+( x i · − x ·· ) 2 ( x ij − x ·· ) 2 = Now sum over i,j. The middle term drops out after summing J ∑ over j , because ( x ij − x i · ) = 0. Hence j = 1 I J I J I J ( x ij − x ·· ) 2 = ( x ij − x i · ) 2 + ( x i · − x ·· ) 2 ∑ ∑ ∑ ∑ ∑ ∑ SST = i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Fundamental Identity: SST = SSTr + SSE = ( x ij − x i · )+( x i · − x ·· ) x ij − x ·· ( x ij − x i · ) 2 + 2 ( x ij − x i · )( x i · − x ·· )+( x i · − x ·· ) 2 ( x ij − x ·· ) 2 = Now sum over i,j. The middle term drops out after summing J ∑ over j , because ( x ij − x i · ) = 0. Hence j = 1 I J I J I J ( x ij − x ·· ) 2 = ( x ij − x i · ) 2 + ( x i · − x ·· ) 2 = SSE + SSTr ∑ ∑ ∑ ∑ ∑ ∑ SST = i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Fundamental Identity: SST = SSTr + SSE = ( x ij − x i · )+( x i · − x ·· ) x ij − x ·· ( x ij − x i · ) 2 + 2 ( x ij − x i · )( x i · − x ·· )+( x i · − x ·· ) 2 ( x ij − x ·· ) 2 = Now sum over i,j. The middle term drops out after summing J ∑ over j , because ( x ij − x i · ) = 0. Hence j = 1 I J I J I J ( x ij − x ·· ) 2 = ( x ij − x i · ) 2 + ( x i · − x ·· ) 2 = SSE + SSTr ∑ ∑ ∑ ∑ ∑ ∑ SST = i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 1. SST measures the total variation of the data. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Fundamental Identity: SST = SSTr + SSE = ( x ij − x i · )+( x i · − x ·· ) x ij − x ·· ( x ij − x i · ) 2 + 2 ( x ij − x i · )( x i · − x ·· )+( x i · − x ·· ) 2 ( x ij − x ·· ) 2 = Now sum over i,j. The middle term drops out after summing J ∑ over j , because ( x ij − x i · ) = 0. Hence j = 1 I J I J I J ( x ij − x ·· ) 2 = ( x ij − x i · ) 2 + ( x i · − x ·· ) 2 = SSE + SSTr ∑ ∑ ∑ ∑ ∑ ∑ SST = i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 1. SST measures the total variation of the data. 2. SSE is the contribution from the variation within the populations/treatment groups. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Fundamental Identity: SST = SSTr + SSE = ( x ij − x i · )+( x i · − x ·· ) x ij − x ·· ( x ij − x i · ) 2 + 2 ( x ij − x i · )( x i · − x ·· )+( x i · − x ·· ) 2 ( x ij − x ·· ) 2 = Now sum over i,j. The middle term drops out after summing J ∑ over j , because ( x ij − x i · ) = 0. Hence j = 1 I J I J I J ( x ij − x ·· ) 2 = ( x ij − x i · ) 2 + ( x i · − x ·· ) 2 = SSE + SSTr ∑ ∑ ∑ ∑ ∑ ∑ SST = i = 1 j = 1 i = 1 j = 1 i = 1 j = 1 1. SST measures the total variation of the data. 2. SSE is the contribution from the variation within the populations/treatment groups. 3. SSTr is the contribution from between the populations/groups. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Example. Perform an ANOVA on the enclosed test data to see if the “true average performances” can be considered equal. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
The Situation Test Statistic Computing the Quantities Example. Perform an ANOVA on the enclosed test data to see if the “true average performances” can be considered equal. Use a significance level of α = 0 . 05 and also compute the p-value. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Single Factor Analysis of Variance (ANOVA)
Recommend
More recommend
