SMHS ANOVA - Revision history

Dinov: /* See Also */

2026-02-10T21:18:20Z

‎See Also

Dinov: /* References */

2026-02-10T21:18:07Z

‎References

Dinov: /* Problem 1 (One‑Way ANOVA) */

2025-12-05T22:23:25Z

‎Problem 1 (One‑Way ANOVA)

Dinov: /* Problem 1 (One‑Way ANOVA) */

2025-12-05T22:22:57Z

‎Problem 1 (One‑Way ANOVA)

Dinov at 22:22, 5 December 2025

2025-12-05T22:22:01Z

Dinov: /* Applications */

2025-12-05T22:21:06Z

‎Applications

Dinov: /* One‑Way ANOVA */

2025-12-05T22:18:30Z

‎One‑Way ANOVA

Dinov at 22:17, 5 December 2025

2025-12-05T22:17:36Z

Show changes

Dinov: /* Two-way ANOVA */

2014-10-08T21:29:27Z

‎Two-way ANOVA

Dinov: /* Two-way ANOVA */

2014-10-08T21:28:39Z

‎Two-way ANOVA

@@ Line 328: / Line 328: @@
 <hr>
-* SOCR Home page: http://www.socr.umich.edu
+* SOCR Home page: https://socr.umich.edu
-{{translate|pageName=http://wiki.socr.umich.edu/index.php?title=SMHS_ANOVA}}
+{{translate|pageName=https://wiki.socr.umich.edu/index.php?title=SMHS_ANOVA}}

@@ Line 313: / Line 313: @@
 ===References===
 * [http://wiki.stat.ucla.edu/socr/index.php/Probability_and_statistics_EBook#Chapter_XI:_Analysis_of_Variance_.28ANOVA.29 SOCR ANOVA Chapter]
 * [https://en.wikipedia.org/wiki/Analysis_of_variance ANOVA – Wikipedia]

@@ Line 280: / Line 280: @@
 . State hypotheses: <math>H_0: \mu_1 = \mu_2 = \mu_3</math> vs. <math>H_a</math>: at least one mean differs.
 . Check assumptions (normality, equal variance).
 . Compute ANOVA table.
 . Make decision based on F‑test.
 . If significant, perform post‑hoc comparisons.

@@ Line 277: / Line 277: @@
 At <math>\alpha=0.05</math>, test whether the average assembly times differ across brands.
-*Solution outline*:
+* Solution outline:
 . State hypotheses: <math>H_0: \mu_1 = \mu_2 = \mu_3</math> vs. <math>H_a</math>: at least one mean differs.
 . Check assumptions (normality, equal variance).

@@ Line 221: / Line 221: @@
 ====R Commands for ANOVA====
-* One‑way ANOVA: `aov(y ~ group, data)`
+* One‑way ANOVA: aov(y ~ group, data)
-* Two‑way ANOVA (with interaction): `aov(y ~ factor1 * factor2, data)`
+* Two‑way ANOVA (with interaction): aov(y ~ factor1 * factor2, data)
 * Check assumptions:
-   - Normality: `shapiro.test(residuals(model))`
+   - Normality: shapiro.test(residuals(model))
-   - Homogeneity of variances: `leveneTest(y ~ group, data)`
+   - Homogeneity of variances: leveneTest(y ~ group, data)
-* Post‑hoc tests: `TukeyHSD(model)`, `pairwise.t.test(y, group, p.adjust="bonferroni")`
+* Post‑hoc tests: TukeyHSD(model), pairwise.t.test(y, group, p.adjust="bonferroni")
-* Effect size: `library(effectsize); eta_squared(model)`
+* Effect size: library(effectsize); eta_squared(model)
 ====Complete R Example (One‑Way)====

@@ Line 190: / Line 190: @@
 </pre>
-#### Example 2: Two‑Way ANOVA with Interaction
+==== Example 2: Two‑Way ANOVA with Interaction====
 A study examines how gender (M/F) and exercise regimen (None, Light, Heavy) affect cholesterol level (mg/dL). Data are balanced with 5 subjects per cell.

@@ Line 34: / Line 34: @@
 * <math>y_{ij}</math> = <math>j</math>‑th observation in group <math>i</math> (<math>i=1,\dots,k</math>; <math>j=1,\dots,n_i</math>).
-* <math>n_i</math> = number of observations in group <math>i</math>.
+* <math>n_i</math> = number of observations in group <math>i.</math>
 * <math>n = \sum_{i=1}^k n_i</math> = total sample size.
-* Group mean: <math>\bar{y}_{i.} = \frac{1}{n_i}\sum_{j=1}^{n_i} y_{ij}</math>.
+* Group mean: <math>\bar{y}_{i.} = \frac{1}{n_i}\sum_{j=1}^{n_i} y_{ij}.</math>
-* Grand mean: <math>\bar{y}_{..} = \frac{1}{n}\sum_{i=1}^k \sum_{j=1}^{n_i} y_{ij}</math>.
+* Grand mean: <math>\bar{y}_{..} = \frac{1}{n}\sum_{i=1}^k \sum_{j=1}^{n_i} y_{ij}.</math>
 *Variance decomposition:
-* Total sum of squares: <math>SS_{total} = \sum_{i=1}^k \sum_{j=1}^{n_i} (y_{ij} - \bar{y}_{..})^2</math> with <math>df_{total}=n-1</math>.
+* Total sum of squares: <math>SS_{total} = \sum_{i=1}^k \sum_{j=1}^{n_i} (y_{ij} - \bar{y}_{..})^2</math> with <math>df_{total}=n-1.</math>
-* Between‑groups sum of squares: <math>SS_{between} = \sum_{i=1}^k n_i (\bar{y}_{i.} - \bar{y}_{..})^2</math> with <math>df_{between}=k-1</math>.
+* Between‑groups sum of squares: <math>SS_{between} = \sum_{i=1}^k n_i (\bar{y}_{i.} - \bar{y}_{..})^2</math> with <math>df_{between}=k-1.</math>
-* Within‑groups (error) sum of squares: <math>SS_{within} = \sum_{i=1}^k \sum_{j=1}^{n_i} (y_{ij} - \bar{y}_{i.})^2</math> with <math>df_{within}=n-k</math>.
+* Within‑groups (error) sum of squares: <math>SS_{within} = \sum_{i=1}^k \sum_{j=1}^{n_i} (y_{ij} - \bar{y}_{i.})^2</math> with <math>df_{within}=n-k.</math>
 The fundamental identity is:
-<math>SS_{total} = SS_{between} + SS_{within}</math>, and likewise <math>df_{total} = df_{between} + df_{within}</math>.
+<math>SS_{total} = SS_{between} + SS_{within}</math>, and likewise <math>df_{total} = df_{between} + df_{within}.</math>
 *Mean squares and F‑statistic:
 * <math>MS_{between} = \frac{SS_{between}}{df_{between}}</math>
 * <math>MS_{within} = \frac{SS_{within}}{df_{within}}</math>
-* Test statistic: <math>F = \frac{MS_{between}}{MS_{within}}</math>
+* Test statistic: <math>F = \frac{MS_{between}}{MS_{within}}.</math>
 Under <math>H_0</math> (all population means equal), <math>F</math> follows an F‑distribution with <math>(k-1,\; n-k)</math> degrees of freedom. A large <math>F</math> value suggests that the between‑group variation is substantial relative to within‑group variation, providing evidence against <math>H_0</math>.

@@ Line 161: / Line 161: @@
 :: (4) the groups have the same sample size.
-* Studying the residuals: When the ANOVA assumptions hold, the residuals should be approximately normally distributed. Otherwise, there may be outliers in the data, one or more groups may come from non-normal distributions, or the linear mode is just not able to capture the data complexity. Normality testing of the residuals gives an indication of whether the sample populations are roughly normal, but may not indicate the underlying cause of this non-normality:
+* Studying the residuals: When the ANOVA assumptions hold, the residuals should be approximately normally distributed. Otherwise, there may be outliers in the data, one or more groups may come from non-normal distributions, or the linear mode is just not able to capture the data complexity. Normality testing of the residuals gives an indication of whether the sample populations are roughly normal, but may not indicate the underlying cause of this non-normality. Normality tests include:
 ** [[SOCR_EduMaterials_AnalysisActivities_KolmogorovSmirnoff|Kolmogorov-Smirnov test]] and [http://socr.umich.edu/html/dist Anderson-Darling test] are based on the estimate of the empirical CDF.
 ** [[AP_Statistics_Curriculum_2007_MultiVar_ANOVA#Wilk.E2.80.99s_.5C.28.5CLambda.5C.29 | Shapiro-Wilk’s test]] is a correlation based test applicable for $n < 50$.

@@ Line 136: / Line 136: @@
 **Interaction effect: the effect that one factor has on the other factor. The degree of freedom is the product of the two degrees of freedom of each factor.
-*Calculations:
+*Calculations: It is assumed that main effect A has $a$ levels (and $df(A) = a-1$), main effect B has $b$ levels (and ($df(B) = b-1$), $r$ is the sample size of each treatment, and $N = a\times b\times r$ is the total sample size. Notice the overall degree of freedom is once again one less than the total sample size.
 <center>
@@ Line 162: / Line 161: @@
 :: (4) the groups have the same sample size.
-* Studying the residuals: When the ANOVA assumptions hold, the residuals should be approximately normally distributed. Otherwise, there may be outliers in the
+* Studying the residuals: When the ANOVA assumptions hold, the residuals should be approximately normally distributed. Otherwise, there may be outliers in the data, one or more groups may come from non-normal distributions, or the linear mode is just not able to capture the data complexity. Normality testing of the residuals gives an indication of whether the sample populations are roughly normal, but may not indicate the underlying cause of this non-normality:
 Normality tests:
 ** [[SOCR_EduMaterials_AnalysisActivities_KolmogorovSmirnoff|Kolmogorov-Smirnov test]] and [http://socr.umich.edu/html/dist Anderson-Darling test] are based on the estimate of the empirical CDF.