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ANOVA: Part II

ANOVA: Part II. Last week. Introduced to a new test: One-Way ANOVA ANOVA’s are used to minimize family-wise error: If the ANOVA is statistically significant: One or more groups are significantly different in your test Follow-up testing required to determine which groups

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ANOVA: Part II

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  1. ANOVA: Part II

  2. Last week • Introduced to a new test: • One-Way ANOVA • ANOVA’s are used to minimize family-wise error: • If the ANOVA is statistically significant: • One or more groups are significantly different in your test • Follow-up testing required to determine which groups • If the ANOVA is not statistically significant: • All groups are statistically similar (can’t be confident enough that any differences aren’t simply random sampling error) • No follow-up testing is needed, all will also be non-significant These ‘rules’ are true for all ANOVA’s

  3. Tonight… • We’ll finish testing for ‘group differences’ with: • Factorial ANOVA • Any ANOVA including more than one IV • Introduce interactions • ‘Hint at’: • ANCOVA • ANOVA’s including a co-variate • Repeated Measures ANOVA • ANOVA’s for dependent, paired, or related groups

  4. Something to think about… • One-Way ANOVA’s ‘look’ very similarly to t-tests • They compare individual groups, one by one • Compare means between the levels of one variable • Factorial ANOVA’s, ANCOVA’s, and RM ANOVA’s are actually more closely related to multiple linear regression • All of these tests use the General Linear Model • Y = mX + b • We will be comparing groups, but we’ll be doing it using the General Linear Model behind the scenes • This should make more sense later in the lecture…

  5. Factorial ANOVA example • Example dataset contains patients that were diagnosed with breast cancer • Key Variables: • Survival Time in Months – Survival time after diagnosis/treatment (our dependent variable) • Tumor Size – Subjects grouped by initial tumor size (< 2 cm, 2-5 cm, or > 5 cm, our independent variable) • We want to know if Tumor Size effects the cancer Survival Time

  6. Picture of our analysis • If we ran a One-way ANOVA, it would look something like this: • Will compare mean survival time between each of the three groups and determine if they are significantly different

  7. However… • This is a good analytical plan – but in reality more variables influence cancer survival than just tumor size • We would like to examine several different IV’s • For example, lymph nodes are essentially a means for cancer to travel around the body. If the cancer spreads, survival time decreases • Lymph node involvement may also influence survival time • Instead of 1 – we could run 2 one-way ANOVA’s…

  8. New Plan • ANOVA #1: • ANOVA #2: This will tell us if Tumor Size impacts survival This will tell us Lymph Node involvement impacts survival

  9. But… • Now our two tests examine how two independent variables influence our dependent variable but… • …what if Tumor Size and Lymph Node involvement are related to each other? • Remember collinearity? This is a similar concept • Called Interaction in ANOVA (more on this later) • By running two separate tests we are ignoring this potential interaction of the variables

  10. Factorial ANOVA’s • Running a factorial ANOVA allows us to run one ANOVA that: • 1) Uses two (or more) independent variables • 2) Examines the potential for interaction between our IV’s • Again, notice here that Factorial ANOVA’s are very similar to multiple regression • We’re using two or more independent variables and one dependent variable • The only difference is the Factorial ANOVA tests for group differences while the Regressionpredicts

  11. Terminology • In Factorial ANOVA’s the independent variables are called ‘main effects’ • Tumor Size is a main effect • Lymph Node involvement is a main effect • This is just the lingo, think of it as ‘main’ variable • Factorial ANOVA’s are usually described by how many levels each main effect has • How many levels does our Tumor Size variable have? • How many levels does our Lymph Node variable have? • We have a 3 x 2 Factorial ANOVA

  12. New Analysis: 6 Groups Main Effect #1 • Instead of two tests – everything is tested at once • Let’s run this model in SPSS… Two-Way ANOVA Or 3 X 2 Factorial ANOVA Main Effect #2

  13. Factorial ANOVA use the General Linear Model tab • The “Univariate” description refers to the number of dependent variables you have – not independent • Again you can see a clear separation of One-Way ANOVA’s (that are like t-tests) and all the other ANOVA’s (that are like multiple regression). SPSS puts them under different options.

  14. Dependent Variable is Survival Time • Fixed Factors = IV’s • Tumor Size • Lymph Node • Post-Hoc Tests same as before • Options similar (can get means, SD, for each group, etc…)

  15. For Post-Hoc options now I can select all of my independent variables (not just 1)

  16. Use the “Plots” option every time you run a Factorial ANOVA – we’ll discuss why later • It really doesn’t matter which IV you enter first, but I tend to “Separate Lines” by the IV with the fewest levels, in this case Lymph Nodes

  17. Estimated Marginal Means • Factorial ANOVA’s have another new addition under the ‘Options’ button • The Post-Hoc tab will use Pairwise Comparisions to compare the levels of our independent variables (Tumor Size and Lymph Node) and look for differences • However, the Estimated Marginal Means are required when you want to make comparisons using BOTH of your two main effects at the same time • Lets use our picture…

  18. What Post-Hoc Tests do… That is NOT the only thing we are interested in! • Post-Hoc tests will compare Tumor Size between ONLY these three groups (ignoring the effect of Lymph Node Involvement) • And the post-hoc test will compare Lymph Nodes, ignoring Tumor Size

  19. Estimated Marginal Means vs Post-Hoc • In other words: • Post-Hoc tests: Test for group difference one independent variable at a time • Adequate for One-Way ANOVA’s • Like a t-test • Estimated Marginal Means: Test for group differences while considering all variables in your model • More appropriate for Factorial ANOVA’s

  20. How do they Work? • This option will ‘adjust’ the group means of Tumor Size for Lymph Node Involvement • If everyone had the same amount of lymph node involvement – what effect would Tumor Size have? • These are ‘estimated’ means because they do NOT exist in real life – SPSS is saying, “What if…?” X1 X2 X3

  21. Estimated Marginal Means • These are ‘estimated’ means because they do NOT exist in real life – SPSS is saying, “What if…?” • Has SPSS lost it’s mind? • How can we ‘adjust’ for another variable? • How can we ‘estimate’ how an independent variable will effect the dependent variable? • SPSS is using the General Linear Model to ‘predict’

  22. The General Linear Model in ANOVA • What SPSS is really doing: • Survival Time = TumorSizeX1 + LymphX2 + b • Estimated Marginal Means function by assuming a constant value for one of the X’s – in this case it is Lymph Node Involvement: • Survival Time1 = (<2cm)X1 + (Constant)X2 + b • Survival Time 2 = (2-5cm)X1 + (Constant)X2 + b • Survival Time 3 = (>5cm)X1 + (Constant)X2 + b • The constant for the IV is similar to the mean lymph node involvement for the sample

  23. Estimated Marginal Means • This option will ‘adjust’ the group means of Tumor Size for Lymph Node Involvement • If everyone had the same amount of lymph node involvement – what effect would Tumor Size have? Estimated Survival Time for everyone with a Tumor Size < 2cm (adjusting for Lymph Node Involvement) Initial Questions on Estimated Marginal Means? X1 X2 X3

  24. Move over all factors into the box on the right • Display Estimated Means for… • Check the box that says “Compare Main Effects” • This will generate p-values for the groups – adjusted for the other variables

  25. Factorial Outputs • Once you run the test the Output file can be overwhelming since there is so much to look at. • Move through one output box at a time • For this 1 analysis, I requested: • Descriptive statistics • A Means plot • Estimated Marginal Means for Tumor Size, Lymph Node Involvement, and our Interaction • Post-Hoc tests and pair-wise comparisons • …And the actual ANOVA

  26. Ready for the results…?

  27. Sample Sizes • You want to try and have all groups be equal in size, but it’s almost impossible to do! • How many subjects in our study? 826 + 274 + 42= 1142= 871+ 271

  28. Descriptives for all 6 groups Main Effect #1: Tumor Size Main Effect #2 Lymph Node?

  29. ANOVA Results • Take note of df, F, and Significance for each Main Effect (we have 2) and each Interaction (we have 1) • Also, you will report the Error (or ‘overall’)df = 1136

  30. Initial interpretation: • Survival Time is significantly impacted by Tumor Size • p < 0.001 • Survival Time is not significantly impacted by Lymph Node Involvement • p = 0.207 • Survival Time is significantly impacted by the interaction of Tumor Size and Lymph Node Involvement • p = 0.016 Next Step? Estimated Marginal Means to determine WHERE the differences are!

  31. EMM for Tumor Size • Estimated Marginal Means/Pairwise Comparisons for Tumor Size (adjusting for Lymph Node) • Looks just like a Post-Hoc test – except it’s based on the Estimated Marginal Means!

  32. EMM for Tumor Size • Interpretation: • Those with Tumors <= 2 cm tended to survive longer than those in the 2-5 cm group (p = 0.001) and > 5 cm group (p = 0.006). There was not a significant difference between the 2-5 cm and > 5 cm group (p = 0.254).

  33. ? • Do I need to look at the Estimated Marginal Means for Lymph Node involvement? • Was the Main Effect of Lymph Node statistically significant? • NO! But you can verify that for yourself:

  34. Interactions • Aside from the Main Effects, we did have a statistically significant interaction we need to consider • Interactions: • A combination of Main Effects. Indicates that two IV’s are related to each other • Similar to: collinearity, confounders, etc… • If you have a significant interaction: • At least one of your Main Effects is influenced by another main effect

  35. Interactions • Think of it this way… • When considered alone: • As tumor size increased survival time decreased • Lymph node involvement had no effect on survival time • An statistically significant interaction means… • The effect that tumor size has on survival time is MODIFIED by lymph node involvement • Interactions also known as ‘effect modification’ • The two variables interact to cause a different effect on survival • The effect of tumor size is dependent on lymph node involvement!

  36. Interactions • When you have a statistically significant interaction, you have to ignore any statistically significant main effects • Why? Because the interaction ‘overrides’ them • So, in writing you would report the significant main effect, but realize the interaction is MORE important • This is why we requested the means plot earlier. A plot of the means (including both the interaction terms) makes it easier to see…

  37. You can see clearly that Lymph Node does have an effect – but only for those with tumors larger than 2 cm • If there is no interaction, these lines would be parallel • The effect of increasing tumor size would be the same for both lines No difference Big difference Big difference

  38. Interactions • In general, if the lines on your plot are NOT parallel, you probably have a significant interaction • Without effect modification/interactions, you would expect the variables to have the same impact no matter what their combination • Examples…

  39. Example: Interaction

  40. Example: NO Interaction

  41. Example: Interaction

  42. The interaction is significant…and we can see it on the plot…what now…?

  43. Significant Interactions • In truth, it’s much easier if you do NOT find significant interactions • Report only the statistically significant main effects • Use the Post-Hoc tests or Estimated Marginal Means to see which groups are different • Ignore the non-significant main effects • Ignore the non-significant interaction • When you do find a significant interaction, you must now treat each group separately (in our case, consider all 6 groups)

  44. Estimated Marginal Means Interaction Table • When you request it, you will get a pairwise comparison table for the interaction using the estimated marginal means • You have to use the estimated marginal means here because we have ‘combined’ two variables • This table will tell you which groups are significantly different – but you have to use the 95% confidence intervals!

  45. Estimated Marginal Means Interaction • When 95% confidence intervals for 2 groups do NOT overlap, they are significantly different (p < 0.05) • Which groups are statistically different here? • Compare the two groups WITHIN each category of Tumor Size

  46. Reporting Your Findings • A 3 x 2 Factorial ANOVA revealed a statistically significant main effect for Tumor Size (F(2, 1136) = 7.719, p < 0.001). In general, as tumor size increased, survival time decreased*. The main effect of Lymph Node Involvement was not statistically significant (F(1, 1136) = 1.592, p = 0.207). However, the Tumor Size x Lymph Node interaction was statistically significant (F(2, 1136) = 4.139, p = 0.016. For individuals with tumors <2cm, lymph node involvement made no difference. For those in the 2-5cm and >5cm group, lymph node involvement further decreased survival time*. • *You could list the means/SD for each group and reveal the results of the pairwise comparisons. But most of the time this information is place in a table (with 95% CI’s).

  47. Checklist for Factorial ANOVA’s • Look at group descriptives and sample size • Examine the ANOVA output box (Tests of Between Subjects Effects) to see df, F ratio’s, and p-values for the Main Effects and Interactions(s) • If you have significant Main Effects • Use Estimated Marginal Means + Pair-wise Comparisons to see which groups are different (adjusting for the other variables in your model) • If you have a significant Interaction: • Ignore the Main effects! The interaction is what is REALLY going on! • Use the means Plot to see how your Main Effects work together • Use the estimated marginal means interaction table and the 95% confidence intervals to see which groups are different Questions on the Checklist?

  48. Factorial ANOVA’s • Our example here includes a 3-level variable and a 2-level variable. • As you add levels – or variables – these things will really grow quickly • Example: Besides tumor size and lymph node involvement, progesterone receptor status (2-level; positive or negative) might also be relevant to cancer survival. I’ll add it to our ANOVA: • DV = Survival Time • IV = Tumor Size, Lymph Node Involvement, Progesterone Receptor Status • What kind of Factorial ANOVA is this now? 3 x 2 x 2 ANOVA

  49. Note my NEW main effects and interactionS • What would I need to do now?

  50. QUESTIONS on Factorial ANOVA’s?

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