This study used data about obesity, physical activity, injury, age, gender and OA. Descriptive statistics such as mean, range, minimum-maximum and standard deviation (SD) for continuous variables, and the frequency tables (count and percentage) for categorical variables will be presented.
In this study, the outcomes for the incidence and progression of knee OA were binary: participants had incidence of knee OA or not, or participants had progression of knee OA or not. Therefore, a logistic regression model was used to calculate the crude and adjusted ORs and their 95% CIs for the association between knee OA and the predictor variables, namely: obesity and knee OA; physical activity and knee OA; injury and knee OA; age and knee OA; and gender and knee OA at the last follow-up.
Statistical analyses were performed at knee level in this study. Therefore, using the standard logistic regression would have underestimated the standard errors due to the inter-knees correlation in each subject; and consequently, resulted in underestimation of corresponding ORs and incorrect p values. Hence, the logistic regression model was performed using the generalized estimating equation (GEE) with exchangeable correlation matrix to adjust for the correlation between knees
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within subject (Niu, Zhang, LaValley et al. 2003; Haugen, Slatkowsky-Christensen, Bøyesen et al. 2013).
This study also examined the two-way and three-way interactions between, physical activity, obesity and injury on the risk of incidence and progression of radiographic and symptomatic knee OA. Generally, the interaction can be assessed based on additive or multiplicative scales.
The multiplicative measure of interaction has been more frequently reported in epidemiological literature when the outcome was binary. This is because it could be easily obtained from a logistic regression model using the standard statistical software packages. However, assessing the measure of interaction on an additive scale needed extra analysis, which is not readily available in standard statistical software packages. Thus, this could be another reason for more frequent reporting of multiplicative over the additive interaction, rather than a careful thought given on the choice of interaction.
However, there is a general consensus that the measure of interaction on an additive scale is more appropriate for evaluating the interaction in the biological and public health research (Knol and VanderWeele 2012). Therefore, this study
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assessed the measure of interaction on both additive and multiplicative scales.
In this study, the relative excess risk due to interaction (RERI) on the multiplicative scale was obtained by entering the interaction term into the logistic regression model (Figure 2- 1). For instance, to investigate the two-way interaction between physical activity and injury on the risk of knee OA, the model included obesity, injury, physical activity, age and gender as independent variables, knee OA as the outcome measure, plus the two-way interaction term of “physical activity*injury”. Similar models were used to investigate the other two-way and three-way interactions.
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Logistic regression model:
In (OR) = (exposure A+B-) + (exposure A-B+) + (exposure A+B+) + ……..
=In (reference group) = regression coefficient of back
ground risk when both exposure A and B are absent (intercept)
= In (ORA+B-) = regression coefficient of main effect of
exposure A on the outcome when exposure B is absent =In (ORB+A-) = regression coefficient of main effect of
exposure B on the outcome when exposure A is absent =In ((ORA+B+)/ (ORA+B- * ORB+A-))
RERI multiplicative = e = (ORA+B+)/ (ORA+B- * ORB+A-)
Thus, if:
RERI > 1: positive interaction RERI < 1: negative interaction RERI = 1: no interaction
RERI multiplicative-three way interaction = (ORA+B+C+)/ (ORA+B-C- * ORB+A- C- * ORC+A-B- * ORA+B+C- * ORA+C+B- * ORA-B+C+)
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For the additive scale, the RERI was also estimated by using OR values. This was calculated based on the Rothman’s method, a linear OR model of regression (Rothman 1986; Richardson and Kaufman 2009). In this method, all odds values (regression coefficients) used in the linear regression model were turned into OR by dividing them into the background odds of disease (Figure 2-2; Figure 2-3). Thus, the regression coefficient of interaction in the linear regression model was computed based on the OR differences rather than the odds differences. The formula in Figure 2-2 and Figure 2-3 describes how the measure of interaction on the additive scale was derived from the linear regression model using ORs (Knol, van der Tweel, Grobbee et al. 2007).
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Linear regression model:
Y = + (exposure A+B-) + (exposure A-B+) +
(exposure A+B+) + ……
P = odds
= (PA-B-) Regression coefficient of back ground effect
when exposure A and B are both absent (intercept)
= (PA+B- – PA-B-)= Regression coefficient of main effect of
exposure A on the outcome when exposure B is absent
(PB+A- – PA-B-) = Regression coefficient of main effect of
exposure B on the outcome when exposure A is absent
=amount of interaction on the additive scale base on the
odds difference: (PA+B+ - PA-B-) – ((PA+B- - PA-B-) PB+A- - PA-B-)) = PA+B+ – PA+B- PB+A- + PA-B-
Rothman model: (using ORs in the linear regression model)
PA+B+ – PA+B- PB+A- + PA-B- RERIOR = (PA+B+/PA-B-) –
(PA+B-/PA-B-) – (PB+A-/PA-B-) + (PA-B-/PA-B-)
RERI additive (OR) =(ORA+B+)– (ORA+B-)– (ORB+A-) + 1
Thus, if:
RERI > 0: positive interaction
RERI < 0: negative interaction
RERI = 0: no interaction
Delta method was used to estimate the 95% CIs and the corresponding p value for RERI on additive scale (Hosmer and Lemeshow 1992). These were obtained via the nlcom comment in Stata.
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Figure 2-3: The three-way interaction model for the additive
scale
Findings of interactions analyses were presented according to the STROBE recommendations (Knol and VanderWeele 2012), where the separate effects of exposures and their joint effects were reported in addition to the measure of interaction on additive and multiplicative scales. For all interaction analyses, the lowest risk groups were considered as the single reference category. For instance, for the interaction between obesity and injury on the risk of knee OA, participants were divided into the four categories of “obese and injured”, “obese and uninjured”, “non-obese and injured”, and “non-obese and uninjured” groups. The group of “non-obese and uninjured”
Rothman model: (using ORs in the linear regression model) RERI three-way interaction (OR) = (ORA+B+C+)- (ORA+B-C-) – (ORB+A-C-)–
(ORC+A-B-)– (RERIA+B+C-)- (RERIA+C+B-)- (RERIA-B+C+) + 2
Thus, if:
RERI > 0: positive interaction
RERI < 0: negative interaction
RERI = 0: no interaction
In the three way interaction model, the two way interactions between A and B; A and C; B and C were computed as follows:
RERIA+B+C- = (ORA+B+C-)– (ORA+B-C-)– (ORA-B+C-) + 1
RERIA+C+B-= (ORA+C+B-)– (ORA+C-B-)– (ORA-C+B-) + 1
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individuals were considered as the reference category, so the risk of knee OA in the other three groups were compared with the single reference category. These analyses were repeated for the two-way interactions between physical activity and obesity (reference category: inactive-non-obese), and separately for the two-way interaction between physical activity and injury (reference category: inactive-uninjured). For the three-way interaction analyses, data were stratified by physical activity, obesity and injury, in which “non-obese- uninjured-inactive” group was considered as the single reference category. The interaction on multiplicative scale was present if the combined effect of both exposures on the outcome was larger (or smaller) than the multiple of the individual effects of each exposure. For the additive scale, the interaction was present if the combined effect of both exposures on the outcome was larger (or smaller) than the sum of the individual effects of each exposure (Knol et al. 2007).
In this study, all the analyses were performed using Stata version 13 for Windows. In addition, all ORs were adjusted for the confounding effect of age, gender, obesity, injury and physical activity.
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