Practical Considerations for ANOVA Applied Statistics and - PowerPoint PPT Presentation

Model diagnostics Treatment Comparisons Multiple comparisons Practical Considerations for ANOVA Applied Statistics and Experimental Design Chapter 5 Peter Hoff Statistics, Biostatistics and the CSSS University of Washington

Model diagnostics Treatment Comparisons Multiple comparisons Model diagnostics Our model is y ij = µ j + ǫ ij . We have shown that, if A1: { ǫ ij } ’s are independent; A2: Var [ ǫ ij ] = σ 2 for all j ; A3: { ǫ ij } ’s are normally distributed. then F = MST / MSE ∼ F m − 1 , N − m ,λ where λ is the noncentrality parameter. If in addition H 0 : µ i = µ for all i = 1 , . . . , m . then the noncentrality parameter is zero and F = MST / MSE ∼ F m − 1 , N − m . We make these assumptions to • do power calculations when designing a study, • test hypotheses after having gathered the data, and • make confidence intervals comparing the different treatments. What should we do if the model assumptions are not correct?

Model diagnostics Treatment Comparisons Multiple comparisons Model diagnostics Our model is y ij = µ j + ǫ ij . We have shown that, if A1: { ǫ ij } ’s are independent; A2: Var [ ǫ ij ] = σ 2 for all j ; A3: { ǫ ij } ’s are normally distributed. then F = MST / MSE ∼ F m − 1 , N − m ,λ where λ is the noncentrality parameter. If in addition H 0 : µ i = µ for all i = 1 , . . . , m . then the noncentrality parameter is zero and F = MST / MSE ∼ F m − 1 , N − m . We make these assumptions to • do power calculations when designing a study, • test hypotheses after having gathered the data, and • make confidence intervals comparing the different treatments. What should we do if the model assumptions are not correct?

Model diagnostics Treatment Comparisons Multiple comparisons Model diagnostics Our model is y ij = µ j + ǫ ij . We have shown that, if A1: { ǫ ij } ’s are independent; A2: Var [ ǫ ij ] = σ 2 for all j ; A3: { ǫ ij } ’s are normally distributed. then F = MST / MSE ∼ F m − 1 , N − m ,λ where λ is the noncentrality parameter. If in addition H 0 : µ i = µ for all i = 1 , . . . , m . then the noncentrality parameter is zero and F = MST / MSE ∼ F m − 1 , N − m . We make these assumptions to • do power calculations when designing a study, • test hypotheses after having gathered the data, and • make confidence intervals comparing the different treatments. What should we do if the model assumptions are not correct?

Model diagnostics Treatment Comparisons Multiple comparisons Model diagnostics Our model is y ij = µ j + ǫ ij . We have shown that, if A1: { ǫ ij } ’s are independent; A2: Var [ ǫ ij ] = σ 2 for all j ; A3: { ǫ ij } ’s are normally distributed. then F = MST / MSE ∼ F m − 1 , N − m ,λ where λ is the noncentrality parameter. If in addition H 0 : µ i = µ for all i = 1 , . . . , m . then the noncentrality parameter is zero and F = MST / MSE ∼ F m − 1 , N − m . We make these assumptions to • do power calculations when designing a study, • test hypotheses after having gathered the data, and • make confidence intervals comparing the different treatments. What should we do if the model assumptions are not correct?

Model diagnostics Treatment Comparisons Multiple comparisons Model diagnostics Our model is y ij = µ j + ǫ ij . We have shown that, if A1: { ǫ ij } ’s are independent; A2: Var [ ǫ ij ] = σ 2 for all j ; A3: { ǫ ij } ’s are normally distributed. then F = MST / MSE ∼ F m − 1 , N − m ,λ where λ is the noncentrality parameter. If in addition H 0 : µ i = µ for all i = 1 , . . . , m . then the noncentrality parameter is zero and F = MST / MSE ∼ F m − 1 , N − m . We make these assumptions to • do power calculations when designing a study, • test hypotheses after having gathered the data, and • make confidence intervals comparing the different treatments. What should we do if the model assumptions are not correct?

Model diagnostics Treatment Comparisons Multiple comparisons Model diagnostics Our model is y ij = µ j + ǫ ij . We have shown that, if A1: { ǫ ij } ’s are independent; A2: Var [ ǫ ij ] = σ 2 for all j ; A3: { ǫ ij } ’s are normally distributed. then F = MST / MSE ∼ F m − 1 , N − m ,λ where λ is the noncentrality parameter. If in addition H 0 : µ i = µ for all i = 1 , . . . , m . then the noncentrality parameter is zero and F = MST / MSE ∼ F m − 1 , N − m . We make these assumptions to • do power calculations when designing a study, • test hypotheses after having gathered the data, and • make confidence intervals comparing the different treatments. What should we do if the model assumptions are not correct?

Model diagnostics Treatment Comparisons Multiple comparisons Model diagnostics Our model is y ij = µ j + ǫ ij . We have shown that, if A1: { ǫ ij } ’s are independent; A2: Var [ ǫ ij ] = σ 2 for all j ; A3: { ǫ ij } ’s are normally distributed. then F = MST / MSE ∼ F m − 1 , N − m ,λ where λ is the noncentrality parameter. If in addition H 0 : µ i = µ for all i = 1 , . . . , m . then the noncentrality parameter is zero and F = MST / MSE ∼ F m − 1 , N − m . We make these assumptions to • do power calculations when designing a study, • test hypotheses after having gathered the data, and • make confidence intervals comparing the different treatments. What should we do if the model assumptions are not correct?

Model diagnostics Treatment Comparisons Multiple comparisons Model diagnostics Our model is y ij = µ j + ǫ ij . We have shown that, if A1: { ǫ ij } ’s are independent; A2: Var [ ǫ ij ] = σ 2 for all j ; A3: { ǫ ij } ’s are normally distributed. then F = MST / MSE ∼ F m − 1 , N − m ,λ where λ is the noncentrality parameter. If in addition H 0 : µ i = µ for all i = 1 , . . . , m . then the noncentrality parameter is zero and F = MST / MSE ∼ F m − 1 , N − m . We make these assumptions to • do power calculations when designing a study, • test hypotheses after having gathered the data, and • make confidence intervals comparing the different treatments. What should we do if the model assumptions are not correct?

Model diagnostics Treatment Comparisons Multiple comparisons Model diagnostics Our model is y ij = µ j + ǫ ij . We have shown that, if A1: { ǫ ij } ’s are independent; A2: Var [ ǫ ij ] = σ 2 for all j ; A3: { ǫ ij } ’s are normally distributed. then F = MST / MSE ∼ F m − 1 , N − m ,λ where λ is the noncentrality parameter. If in addition H 0 : µ i = µ for all i = 1 , . . . , m . then the noncentrality parameter is zero and F = MST / MSE ∼ F m − 1 , N − m . We make these assumptions to • do power calculations when designing a study, • test hypotheses after having gathered the data, and • make confidence intervals comparing the different treatments. What should we do if the model assumptions are not correct?

Model diagnostics Treatment Comparisons Multiple comparisons Model diagnostics Our model is y ij = µ j + ǫ ij . We have shown that, if A1: { ǫ ij } ’s are independent; A2: Var [ ǫ ij ] = σ 2 for all j ; A3: { ǫ ij } ’s are normally distributed. then F = MST / MSE ∼ F m − 1 , N − m ,λ where λ is the noncentrality parameter. If in addition H 0 : µ i = µ for all i = 1 , . . . , m . then the noncentrality parameter is zero and F = MST / MSE ∼ F m − 1 , N − m . We make these assumptions to • do power calculations when designing a study, • test hypotheses after having gathered the data, and • make confidence intervals comparing the different treatments. What should we do if the model assumptions are not correct?

Model diagnostics Treatment Comparisons Multiple comparisons Residual analysis These assumptions could be more compactly written as A 0 : { ǫ ij } ∼ i.i.d. normal(0 , σ 2 ) However, some assumptions are more important than others. Statistical folklore says the order of importance is A 1, A 2 then A 3. We will discuss A 1 in Chapter 6. For now we will talk about A 2 and A 3.

Model diagnostics Treatment Comparisons Multiple comparisons Residual analysis These assumptions could be more compactly written as A 0 : { ǫ ij } ∼ i.i.d. normal(0 , σ 2 ) However, some assumptions are more important than others. Statistical folklore says the order of importance is A 1, A 2 then A 3. We will discuss A 1 in Chapter 6. For now we will talk about A 2 and A 3.

Model diagnostics Treatment Comparisons Multiple comparisons Residual analysis These assumptions could be more compactly written as A 0 : { ǫ ij } ∼ i.i.d. normal(0 , σ 2 ) However, some assumptions are more important than others. Statistical folklore says the order of importance is A 1, A 2 then A 3. We will discuss A 1 in Chapter 6. For now we will talk about A 2 and A 3.

Model diagnostics Treatment Comparisons Multiple comparisons Residual analysis These assumptions could be more compactly written as A 0 : { ǫ ij } ∼ i.i.d. normal(0 , σ 2 ) However, some assumptions are more important than others. Statistical folklore says the order of importance is A 1, A 2 then A 3. We will discuss A 1 in Chapter 6. For now we will talk about A 2 and A 3.