- To determine if the means differ significantly among the factor groups
- To determine if the means differ significantly among pairs, given that one-way ANOVA finds significant differences among factor groups.

- Your data contain several separate factor groups shown in 2 vectors
- One vector is the observed values; one vector is to mark your values in different factor groups
- The separate factor groups are independent and identically approximately normally distributed
- Each mean of the factor group follows a normal distribution with the same variance and can be compared

**The categories in the Factor Group**

**Descriptive statistics by group**

- The band inside the box is the median
- The box measures the difference between 75th and 25th percentiles
- Outliers will be in red, if existing

- DF
_{Factor}= [number of factor group categories] -1 - DF
_{Residuals}= [number of sample values] - [number of factor group categories] - MS = SS/DF
- F = MS
_{Factor}/ MS_{Residuals} - P Value < 0.05, then the population means are significantly different among factor groups. (Accept alternative hypothesis)
- P Value >= 0.05, then there is NO significant differences in the factor groups. (Accept null hypothesis)

*In this example, smoking groups showed significant, so we could conclude that FEF were significantly different among the 6 groups. *

**When P < 0.05,** if you want to find which pairwise factor groups are significantly different, please continue with **Multiple Comparison**

**Pairwise P Value Table**

- In the matrix, P < 0.05 indicates the statistical significant in the pairs
- In the matrix, P >= 0.05 indicates no statistically significant differences in the pairs

*In this example, we used Bonferroni-Holm method to explore the possible pairs with P < 0.05.
HS was significant different from the other groups;
LS was significantly different from MS and NS;
MS was significantly different from NI and PS;
NI was significantly different from NS.*

- To determine if the means differ significantly among the Factor1 after controlling for Factor2
- To determine if the means differ significantly among the Factor2 after controlling for Factor1
- To determine if the Factor1 and Factor2 have interaction to effect the outcomes
- To determine if the means differ significantly among which pairs, given that two-way ANOVA finds significant differences among groups.

- Your data contain several separate factor groups (or 2 vectors)
- The separate factor groups/sets are independent and identically approximately normally distributed
- Each mean of the factor group follows a normal distribution with the same variance and can be compared

**The categories in the Factor 1**

**The categories in the Factor 2**

- DF
_{Factor}= [number of factor group categories] -1 - DF
_{Interaction}= DF_{Factor1}x DF_{Factor2} - DF
_{Residuals}= [number of sample values] - [number of factor1 group categories] x [number of factor2 group categories] - MS = SS/DF
- F = MS
_{Factor}/ MS_{Residuals} - P Value < 0.05, then the population means are significantly different among factor groups. (Accept alternative hypothesis)
- P Value >= 0.05, then there is NO significant differences in the factor groups. (Accept null hypothesis)

*In this example, dietary types and sex both have effects on the SBP (P<0.001), and dietary types also significantly related with sex (P<0.001). *

**When P < 0.05,** if you want to find which pairwise factor groups are significantly different, please continue with **Multiple Comparison**

**Pairwise P Value Table under Each Factor**

- In the matrix, P < 0.05 indicates the statistical significant in the pairs
- In the matrix, P >= 0.05 indicates no statistically significant differences in the pairs

*In this example, all the pairs, normal vs LV, SV vs LV, SV vs normal, and male vs female had significant differences on SBP.*

- To determine if the means differ significantly among the factor groups
- To determine if the means differ significantly among pairs, given that one-way ANOVA finds significant differences among groups.

- Your data contain several separate factor groups shown in two vectors
- One vector is the observed values; one vector is to mark your values in different factor groups
- The separate factor groups are independent and identically without distribution assumption

**The categories in the Factor Group**

**Descriptive statistics by group**

*In this example, smoking groups showed significant, so we could conclude that FEF were significantly different among the 6 groups from Kruskal-Wallis rank sum test. *

**When P < 0.05,** if you want to find which pairwise factor groups are significantly different, please continue with **Multiple Comparison**

**Reject Null Hypothesis if p <= 0.025**

*In this example, smoking groups showed significant, so we could conclude that FEF were not significantly different in LS-NI, LS-PS, and NI-PS groups. For other groups, P <0.025. *