The quantitative methods employed to analyse the data are described providing an inductive process to interpret the phenomena under investigation (Ritchie and Lewis 2003). Where possible validity of the research methods are provided which is frequently addressed in the literature (Tashakkori and Teddlie 2003). Initially the process for missing data is explained, followed by the factor analysis techniques of exploratory, confirmatory and structural modelling concluded by a summary of the other statistical tests utilized throughout the research.
Missing data
There is no clear guideline regarding what forms a large amount of missing data (Bryne 2010). Little and Rubin (1987) described that there are three patterns of missing data; missing completely at random (MCAR), missing at random (MAR) and non-ignorable missing at random (NMAR). The most popular method to deal with these issues is list wise deletion, which is the fastest and simplest (Bryne 2010), although assumes the data are MCAR (Brown 1994; Arbuckle 2007). Taking this into account observations were deleted if participants had not completed the emotion scale, perceived threat, perceive efficacy or post exposure behavioural smoking scales as practiced in previous research which reduced observations that had a certain
percentage of crucial missing items (Ickes and Sharma 2012). The list wise deleted technique was implemented, reducing the observations from 2237 respondents by 18% to 1837 respondents, although above the industry average amount of 10% it is regarded that each study has different reasons for missing values (Kline 1998). Issues included time restraints, IT equipment and differing levels of participant ability. All the observations with missing values were removed from the data set, while 10 observations were removed from the smoker sample that missed 1 incomplete scale.
Using the pattern matching approach with the mean imputation (Bryne 2010) the 10 observations values were replaced with the mean value per manipulation (n=5). This overcame the problems that would prevent the structural equations modelling to be completed which are not possible when the covariance structure is formed from incomplete data (Bentler and Chou 1987).
Factor Analysis
Factor analysis is made of a number of stages; initially two phases of factor analysis are conducted to ensure each scale is represented by a unique factor reducing any possible cross loadings followed by the structural equations modelling. The phased approach is widely recommended to test the hypothesised model (Manyiwa and Brennan 2012). Factor analysis is frequently used in experimental fields throughout the social science with the prevalent method being for interpreting self-reported questionnaires (Hogarty et al. 2005). Factor analysis uses the common variance;
covariance and communality, that each observed variable shares with the other observed variables (Bryne 2010). To increase the reliability of the analysis sub samples between 20-30% of the data set are extracted for the Exploratory Factor Analysis (EFA) then the remainder 70-80% of the sample is used for the
Confirmatory Factor Analysis (CFA) to validate and amend the initial assumptions (Gerbing and Hamilton 1996). Although applying the EFA findings to CFA should be done with caution (MacCallum et al. 2002; de Winter, Dodou and Wieringa 2009).
Exploratory factor analysis is a useful heuristic strategy for model specification prior to cross validation with confirmatory factor analysis that is shown to provide better research outcomes (Gerbing and Hamilton 1996). The structural equation models are conducted with the complete data set 100% implementing the recommendations from the iterative factor analysis process that proposed reliable scales and factors.
There are two recommendations about the minimum sample size; the absolute number of cases and the subject-to-variable ratio (Velicer and Fava 1998; MacCallum et al.
1999). Although the rule of thumb about minimum sample size is not always valid (MacCallum et al. 2002; Hogarty et al. 2005), it is important to acknowledge advice about the sample size. The factor analysis samples meet the minimum requirements of having at least 100 observations (Comrey and Lee 1992; Hatcher 1994), with all samples randomly selected to contain at least 150 observations that are shown to provide a convergent and reliable solution (Gerbing and Anderson 1985).
Inconclusive recommendations also exist for the sample to variable ratios, the ‘rule of thumb’ ranges from a minimum of 3:1 to 20:1 (Hair et al. 2006). All factor analysis samples had acceptable ratios, with the smoker data set being 4:1 and the non-smoker sample was 11:1. Although concern must be taken when using the guidelines as research into factor analysis sample size and ratio has shown ‘that there was not a minimum level of N or N:p ratio to achieve good factor recovery across conditions examined’ (Hogarty et al. 2005, p.222) showing it is ultimately down to the
researcher preference and circumstances being used as a reference point rather than concrete requirements.
a) Exploratory Factor Analysis
Exploratory factor analysis determines the number of factors that account for the correlations in the R-matrix (Gray and Kinnear 2012). Computed using the Promax rotation on SPSS (v.20.0) to better represent the complexity of the examined variables as constructs in real life are rarely uncorrelated (Harman 1976). This rotation allows the axis to be non-orthogonal and represents correlated and oblique factors (Gray and Kinnear 2012). Once the model was estimated, the process for elimination included:
low communality, low factor loading, cross loading on more than one factors, not loading on any factor, while ensuring at least three items per factor and retaining as many items as possible acknowledging theoretical assumptions about the factor (Velicer and Fava 1998; Costello and Osborne 2005). A factor with fewer than three items is regarded statistically weak and unstable, as the two variables causes’ bias in the factor parameter estimates which nearly vanishes when more than three items are retained (Gerbing and Anderson 1985; Costello and Osborne 2005). Further measures of sampling adequacy and reliability provided support to remove items.
Communalities of 0.4 to 0.7 are common in behavioural or social data (Costello and Osborne 2005) although those lower should be removed. There for items that loaded below 0.3 on the communalities table were removed during an iterative approach to remove items to obtain a reliable pattern matrix. After assessing communalities for sampling adequacy, the factor score coefficients that describe how the item loads on a certain factor were assessed, while taking into consideration Tabachnick and Fidell (2001) recommendation of including items that loaded above 0.30 with loadings in
behavioural or social data being between 0.3 and 0.5 (Hair et al. 1995; de Winter, Dodou and Wieringa 2009).
b) Confirmatory factor analysis
The confirmatory factor analysis assesses the predetermined number of factors and how they load on each factor (Gray and Kinnear 2012), with the objective to determine the adequacy of the model and goodness of fit to the sample data (Bryne 2010). The approach falls into the model-generating classification provided by Joreskog (1993) which is the most common of the three factor analysis approaches (Bryne 2010). The CFA was computed using the AMOS software (v.20.0). If the variables are reliable with strong effects and the model not being overly complex, smaller samples are acceptable (Bollen and Stine 1990). Initially parameter estimates were reviewed then fit indices and residuals outlining model modifications to increase fit and achieve a more parsimonious model. It is imperative to explain why modifications were completed and how it improves the model. Caution must be taken when removing items, especially as ‘when an initial model fits well, it is probably unwise to modify it to achieve even better fit because modifications may simply be fitting small idiosyncratic characteristics of the sample’ (MacCallum, Roznowski and Necowitz 1992; p. 501). Evidence of misfit are captured in the modification indices representing correlated errors which are systematic, rather than random measurement error and may be caused by the items or the respondents (Aish and Joreskog 1990).
The modification index estimates an improvement in overall fit if a correlation path was added (Kline 2011), although Bryne (2010) suggested to correlate the errors, this must be supported by strong substantive and empirical rational (Joreskog 1993), therefore the items with large modification index were removed to reduce the overlap
in item content, which appears when items essentially repeat the same question (Bryne 2010). There are no strict rules about how to alter modification indices, although the greatest indices should be considered first, with iterations conducted one at a time (Raykov and Marcoulides 2010) as a single change can affect other parts of the solution (Joreskog and Sorbom 1996). The modifications to the model was done in an iterative manner removing the greatest index one at a time, with particular attention given to the items with multiple modification indices. The over-determination of factors (factor-to-variable ratio) highlighted by (MacCallum et al. 2002) was assessed, especially for those factors with over 5 items (MacCallum et al. 1999). This was primarily achieved by assessing the factor loadings, which was assessed simultaneously while reviewing the modification indices. Although the factor loading level threshold is dependent on the researcher’s preference (Tabachnick and Fidell 2007), attention was paid to ensure no items were lower than the minimum 0.30, acknowledging that when having 5 or more items per factor it is desirable to load around 0.50 (Costello and Osborne 2005).
Convergent, Discriminant Validity, Reliability and Linearity tests
A two-step procedure provides methods to monitor scale validity (Gerbing and Anderson 1985) that is used throughout both approaches. Composite reliability establishes internal consistency and requires a value close or above a 0.7 threshold (Fornell and Larcker 1981). The average variance explained assesses the convergent validity and represents the percentage of variance in a measure from the hypothesized factor trait (Fornell and Larcker 1981) and requires a value close or above a 0.50 threshold (Hair et al. 2006). Convergent validity is proven if the factor loadings are significant (Hair et al. 2006) and discriminant validity assessed by the average
variance explained requiring a greater variance with its indicators than with other constructs. This is assessed if the average variance explained square root is superior to the estimated squared correlation among each pair of constructs (Fornell and Larcker 1981). The chi-square difference test assesses that when the factors co-vary that the model is a worse fit, using the Yates chi-squared test that estimates an increase of greater than 3.86 per degree of freedom provides adequate model fit (Camilli and Hopkins 1978). The scale reliability is assessed through the Cronbach alpha statistic providing a measure of the internal consistency of a test or scale with different reports stating the acceptable values of alpha, ranging from 0.70 to 0.95 (Tavakol and Dennick 2011). Although the majority suggest minimum value of near 0.7 (Nunnally and Bernstein 1994), alpha’s equal to or greater than 0.6 are acceptable (Murphy and Davidshofer 1988) as the lowest end of the threshold suggests that coefficients of 0.35 or less represent low reliability (Nunnally 1978). Common method variance refers to possible contamination ensuing from the use of a single measurement method: It can exaggerate the apparent association between two constructs measured with the same method (Wiggins, 1973) which often happens to large data sets composed entirely of self-reports (Paulhus and Vazire 2009) or the data came from the same questionnaire.
Harman’s (1976) one factor test checks if any factors accounted for the majority of the covariance among the variables (Podsakoff and Organ 1986). Regressions were done against all factors and confirmed that there are significant linear relationships between all paths expected in the theoretical model further supporting estimating using the structural equations modelling technique.
c) Structural Equations Modelling
Once the EFA and CFA confirm the factors and scale reliability, structural equations modelling (SEM) tests the structural theory. SEM is often based upon a phenomenon or assumptions which enables a hypothesised model to be tested in a simultaneous analysis of the entire system, subject to the goodness of fit indexes the model can argue for the ‘plausibility of postulated relations among variables’ (Bryne 2010; p.3).
Statistical differences were then computed between groups to assess if one group influences the model more than another. The differences between models and factors can be achieved through group difference Z tests and comparing squared multiple correlations. These statistics are similar statistic to R square value that state even small R square effect can be important (Rosnow and Rosenthal 1989) and acknowledge that values of 0.20 or above are regarded adequate to explain variance (Hair et al. 1995), with the greater the value providing more robust evaluations of the model. Structural equation modelling has been recommended as an approach to examine the effects of coping responses upon attitude and intentional responses.
Tests of Model Fit
Marsh et al. (2004) noted that fit indices have evolved into pseudo hypothesis tests.
Although designed to assess the degree of fit to the data (Barrett 2007), the fit indices used depends on the researcher’s discretion (Hu and Bentler 1999; MacCallum, Browne and Sugawara 1996) as there is no agreed best model fit index (Iacobucci 2010). The model fit indices must be taken with caution and not over emphasised as all the aspects of the model need to be assessed in judgement, factor loading, modification indices, Chi Square and GFI’s. Although the variety of indices, there is
agreement to report the χ2 (and its degrees of freedom and p-value), Goodness of Fit Index (GFI), the Comparative Fit Index (CFI), Tucker- Lewis Index (TLI) and the Root Mean Square Error of Approximation (RMSEA) throughout the literature (Iacobucci 2010). The χ2 is the only inferential statistic that acknowledges significance levels among the model fit indices as the other tests exist as ‘rules-of-thumb’ being descriptive measures (Iacobucci 2010). Kline (2011) suggested that a model demonstrates reasonable fit if the χ2 statistic adjusted by its degrees of freedom does not exceed 3.0 (χ2 / df≤3). It is frequently noted that, values of model fit indices exceeding 0.90 reflect reasonable model–data fit with Hu and Bentler (1998) demonstrating strong performance (power and robustness) of the CFI with it being the index of choice (Bentler 1990). Values representing a well-fitting model are regarded as; GFI >.90, CFI >.90, RMSEA < .06, TLI > .95, and RMR/SRMR < .10/.08 (Bentler 1992; Hu and Bentler 1998; Hu and Bentler 1999), although more demanding cut off values have been proposed that appear to be largely unobtainable in appropriate practice (Marsh, Hau and Wen 2004) showing the need for a holistic view of the model. The indices provide a model fit statement ranging from greater than 0.90 being excellent, to 0.75 being very good, onto good, satisfactory and poor.
Statistical Tests
Alongside factor analysis, inferential statistical analysis techniques are used through-out the pre-tests, pilot tests and final study using SPSS (v.20). Analysis of variance techniques are used to assess the difference between means and to assess the difference between respondents responses classified as high or low response groups with correlations calculated to assess the relationships between the variables
acknowledging correlational relationships that were greater than r=0.50 to be large based on standard estimates of correlation effect sizes (Cohen 1992). Throughout the results the level of significance is classified when significant at the 0.01 level representing**, then significant at the 0.05 level with * and when significant at the less the 0.10 level classified with .10. Mediation analysis in prevention studies is important because the processes that lead to behaviour change can be delineated (MacKinnon 1994), a mediator is an intervening variable (risk/protective factor) that explains (or influences) the desired outcome (Baron and Kenny 1986). Mediation analysis most often guided by the procedures outlined by Baron and Kenny (1986) with the majority of mediation analysis in the psychology research using their procedure making it one of the frequently cited although there are more statistically rigorous methods to assess mediation hypothesis (Preacher and Hayes 2004).
Although proposed by Baron and Kenny (1986) the Sobel test (1982) is rarely used in practice (MacKinnon et al. 2002) as the method described by Baron and Kenny (1986) suffers from a low statistical power (MacKinnon et al. 2002). The alternative approach of bootstrapping the sample is a non-parametric approach to effect size estimation and hypothesis testing that does not make assumptions about the shape of distribution of the variables or the sampling distribution of the statistic (Efron and Tibshirani 1993; Mooney and Duval 1993), providing support to use confidence intervals when assessing the indirect effect of the mediator as formal significance tests of indirect effects are rarely conducted. Although the terms mediated and indirect effects are used interchangeably they are distinctly different as a mediated effect is usually thought of as the special case of indirect effects when there is only one intervening variable (Preacher and Hayes 2008). In order to tests the mediation Preacher and Hayes (2008) ‘indirect macro’ estimated through SPSS (v.20.0).