Figure 1 below show mean values (given by height of bar) of arterial stiffness by age (3 groups; young, middle-aged, older) and amount of light physical activity (PA) groups (2 groups; high, low). The result is obtained from a recent study (Gando et al, Hypertension, 2012) that examined associations between arterial stiffness and amount of light physical activity in 538 healthy adults (men and women).

Arterial stiffness is assumed to be a risk factor for cardiovascular disease, and is in this study measured by carotid-femoral pulse wave velocity (cfPWV; in cm/s; increasing values means increasing arterial stiffness).

Duration and intensity of physical activity (PA) was assessed by a triaxial accelerometer, something that registers movements (acceleration) in different directions. Data were recalculated into intensity levels. A previous validation study showed high correlation with measurements of oxygen uptake (VO_{2}). The device was worn at day-time during a 14-day period. Daily time (minutes) spent with light physical activity was recorded for each person (continuous scale), and categorizing into high/low PA group were defined according to whether a person had value higher or lower than median value for persons at the same age and sex.

- A two-way analysis of variance (ANOVA) was applied to examine whether mean level of arterial stiffness differed by age and/or amount of light physical activity.

According to Figure 1

- does it seem to be any association between age and arterial stiffness? – does it seem to be any association between amount of light PA and arterial stiffness?

(give a short explanation for your answers)

- Two alternative ANOVA models can be defined to examine, and formally test, associations between age, light physical activity and arterial stiffness. – define the two alternative models in terms of parameters to be tested – give a short description of what the different parameters in the model represent (interpretation of it).
- describe the difference in interpretation of results from the two different models.

- How can you choose between the two alternative models, and what do you think would be the most suitable model in this case?
- What is the assumption(s) of the ANOVA model (in general).
- What can you do if the assumptions on the model are not met?

In the same study (Gando et al, 2012), a simple linear regression analyses was performed for each age group to examine association between amount of light physical activity (minutes per day, continuous scale) and arterial stiffness (measured by cfPWV, cm/s) according to age.

Figure 2 below show a scatter-plot between the variables in the regression model (amount of light physical activity and arterial stiffness). A significant association was found, but only among the oldest (> 60 years). Results from the linear regression analysis (estimated regression coefficients, regression line), together with Pearson’s correlation coefficient (r), is shown for this group.

- How do you interpret the estimated values of the (unstandardized) regression coefficients (describe in terms of variables in the model)?
- Is it a strong linear association between amount of light physical activity and arterial stiffness among the oldest? Give an explanation for your answer.
- What are the assumptions on the linear regression model? – describe in terms of the variables in the model.
- A multiple linear regression analysis was performed to evaluate whether gender was a potential confounder for the association between amount of light physical activity and arterial stiffness.

Gender did not seem to be a confounder.

– how could the author come to this conclusion, based on results from the linear regression analysis?

In another study, the main aim was to evaluate whether prognosis of colorectal cancer differed between men and women. Kaplan-Meier survival plot and Cox proportional hazard (PH) regression model was applied. The study comprised a total of 247 patients (105 men and 142 women).

The data analysis (Kaplan-Meier method) gave the following results (selected parts of SPSS-output):

**Case Processing Summary**

Gen | der | Total N | N of Events | Censored | |

N | Percent | ||||

Men | 105 | 48 | 57 | 54.3 % | |

Wom | en | 142 | 48 | 94 | 66.2 % |

Over | all | 247 | 96 | 151 | 61.1 % |

**Overall Comparisons**

Chi-Square | Df | Sig. | ||

Log Rank (Mantel-Cox) | 2.276 | 1 | .131 |

Men |

Women |

Test of equality of survival distributions for the different levels of Gender.

- Based on the Kaplan-Meier survival plot
- what is the estimated 50% (median) survival time (approximately) in men and women, respectively
- what is the estimated 75% survival time (approximately) in men and women, respectively
- what is the estimated proportion of survivors (approximately) after 2.5 years (30 months) in men and women, respectively

- Specify the null hypothesis and the alternative hypothesis for the log-rank test.
- Describe result from the survival analysis, based on information in the SPSS output above.

The colorectal cancer data was further analysed in a Cox proportional hazard (PH) regression model.

The SPSS output below show results from an unadjusted analyses (corresponding to Kaplan-Meier survival plot).

**Categorical Variable Codings ^{b}**

Frequency | (1) | ||

Gender^{a} |
1=Men | 105 | 1 |

2=Women | 142 | 0 |

- Indicator Parameter Coding
- Category variable: Gender

**Variables in the Equation**

B | SE | Wald | Df | Sig. | Exp(B) | 95,0% CI for Exp(B) | |||

Lower | Upper | ||||||||

Gender | .305 | .204 | 2.225 | 1 | .136 | 1.356 | .909 | 2.023 |

An analysis adjusted for age at diagnosis (1-year interval, continuous variable) and stage of disease (Duke’s classification system, categorical variable), gave the following result (SPSS output, Model 1)

**Variables in the Equation**

B | SE | Wald | Df | Sig. | Exp(B) | 95,0% CI for Exp(B) | |||

Lower | Upper | ||||||||

Gender | .434 | .209 | 4.326 | 1 | .038 | 1.544 | 1.025 | 2.325 | |

Age_diag | .001 | .009 | .009 | 1 | .922 | 1.001 | .983 | 1.019 | |

Duke | 1 40.374 | 4 | .000 | ||||||

duke( 1) | -2.570 | .472 | 29.603 | 1 | .000 | .077 | .030 | .193 | |

duke( 2) | -2.078 | .343 | 36.771 | 1 | .000 | .125 | .064 | .245 | |

duke( 3) | -.843 | .320 | 6.939 | 1 | .008 | .430 | .230 | .806 | |

duke(4) | 1.636 | .320 | 26.066 | 1 | .000 | 5.135 | 2.740 | 9.624 |

- Describe results (for gender only) from the Cox PH regression analyses, based on information in the SPSS output. Report

– estimated value (point- and interval estimate) of hazard ratio – results from the statistical test

- What is meant by (implies) the proportionality assumption in the Cox PH regression model?
- How can you check whether the proportionality assumption is met?

Additional analyses with age included as a categorical variable (10-yr intervals, <60, 60-69, 70-79, 80+ years; youngest patients as reference group) was also performed. This model gave the following result (SPSS output, Model 2)

**Variables in the Equation**

B | SE | Wald | Df | Sig. | Exp(B) | 95,0% CI for Exp(B) | |||

Lower | Upper | ||||||||

Gender | .407 | .212 | 3.699 | 1 | .054 | 1.502 | .992 | 2.275 | |

Duke | 138.384 | 4 | .000 | ||||||

duke(1) | -2.655 | .474 | 31.379 | 1 | .000 | .070 | .028 | .178 | |

duke(2) | -2.234 | .352 | 40.210 | 1 | .000 | .107 | .054 | .214 | |

duke(3) | -.989 | .327 | 9.123 | 1 | .003 | .372 | .196 | .707 | |

duke(4) | 1.562 | .330 | 22.410 | 1 | .000 | 4.770 | 2.498 | 9.108 | |

Age_10yr | 5.687 | 3 | .128 | ||||||

Age_10yr(1) | -.171 | .285 | .359 | 1 | .549 | .843 | .482 | 1.474 | |

Age_10yr(2) | -.328 | .281 | 1.358 | 1 | .244 | .721 | .415 | 1.250 | |

Age_10yr(3) | .446 | .338 | 1.744 | 1 | .187 | 1.562 | .806 | 3.027 |

- How do you interpret the value of the hazard ratio(s) for age in Model 1 (age as continuous variable) and Model 2 (age as categorical variable), respectively?
- What do you think would be the best model in this case; age included as a continuous variable (1-year interval, Model 1) or as a categorical variable (10-year interval, Model 2)? Give an explanation for your answer.

In the exercise above, assume that it is necessary to adjust for age, despite the lack of significant prognostic impact, and assume that the proportionality assumption is met for all variables in the model.

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