Sample size calculation

Sample size calculations are necessary to ensure that analyses are adequately powered. Power refers to the probability that the proposed analyses will correctly distinguish between the null and experimental hypotheses (Rossi, 1990). A null hypothesis is always one of equilibrium, in other words, that there will be no effect found (where effect could be an association, difference, regression coefficient etc.) (Field, 2009). Where a null hypothesis is wrongly rejected, or where an effect is concluded to exist where it actually doesn't, it is said that a Type I error has been committed; a Type II error is one where a significant effect is concluded, where in feet, this isn't the case (Rosnow & Rosenthal, 1989; Loftus, 1996). The Greek letter a is used to represent the probability of a Type I error occurring and the letter p is used to represent the probability of a Type II error occurring. Within psychology, standard levels of a=.05 and p=.20 are used. Power calculations are more strongly geared to minimisation of a Type II error, therefore, a power level of .80 (1- p) is - considered acceptable. (Loftus, 1996). Power is determined by three factors:

I sample size, effect size, and alpha level (Rossi, 1990). Given that a and p are J known entities, estimates of effect sizes from previous research can be used to hV calculate the required sample size for new research (Wilkinson, 1999). Sample

I size calculations are typically based on the analysis requiring highest sample size 4 pertinent to the primary outcome variables.

In this study, each of the three outcomes (anxiety, depression and quality of

? ttfe) were given equal weighting in terms of importance but the theory testing element was considered secondary to this aim. As regression analyses were

planned for this question, study sample size calculations were based on the minimum number of participants required to sufficiently power multivariate linear regressions. It was estimated that fewer than 20 variables would be entered into each analysis after prior selection based on correlation data (for more detail, see section 3.6.3). External advice on sample size calculations and the proposed analyses was sought from both statistical and health psychology experts.

Simultaneous entry of all assessed variables into a regression would be impossible due to collinearity and potential loss of power. In two cases (emotions and coping), the use of grouped variables were planned; although composite scores lose depth of data (as reported in the systematic review), in this case it seemed justifiable as neither grouped variable was a primary outcome variable.

Rather, with regard to coping, instead of using scores from each coping subscale, the larger maladaptive and adaptive categories were used (perhaps not an ideal scenario (see section 1.5) but a necessary compromise). Similarly, rather than considering each emotional reaction, grouped totals of positive and negative emotion were used. Initial analysis was proposed to consist of a number of correlation analyses between all predictor and outcome variables. Those

variables found to be significantly related with outcome (at p<.01) were then to be altered into regression models (see section 4.5.1.3 for further details of the

statistical plan). Although minimising the number of variables to be entered into regression is not the most statistically ideal method (Cohen 1988, cited in Green 1991), it is a pragmatic method often used within psychological research

(Tabachnick & Fidell, 2000). Where data collection from large samples is not possible, to enter all possible predictors into regressions, whilst being ideal, would be pointless as regression models would become too saturated and would simply faiL Therefore, some statistical compromise is necessary whilst continuing to address substantial research questions in a meaningful and contributory way.

There is a tradition within the behavioural sciences to use standard 'rules of thumb' for estimates of sample sizes required for regression analyses, where average ratios of five data cases to one variable are required (see for example, / Gravetter & Wallnau, 2000; Maxwell, 2000; Tabachnick & Fidell, 2000). Many

other rules of thumb have been suggested, some as high as a ratio of 25:1 (Schmidt, 1971, cited in Green, 1991). Green (1991), however, criticises these

methods of sample size calculation because they fail to take into account

anticipated effect sizes. In this same paper, Green provides the reader with a list of power analysis calculations for various regression scenarios of differing levels of effect size, and various numbers of predictor variables. The calculations are based on Cohen's statistical tables and in his paper, Green compares these to standard rule of thumb calculations. Green provides an unequivocal argument for superior use of power analysis based sample size calculations.

Due to the proposed method of pre-selection of predictor variables, the number, and specific nature, of predictor variables relevant to the regressions was unknown until analysis began. As an a priori estimation, however, the total

number were estimated to range between 15-20 variables. Previous research (see Chapter two) typically reports some small, but usually medium effect sizes for these type of predictor variables and it was not expected that this study would differ substantially from this. Therefore, using Cohen's power based calculations (1988, cited in Green, 1991, and Field, 2009), it was deduced that to find a medium effect size (R2 =0.13), at a power level of .80 (80% chance of detecting an effect within the sample if one truly exists), between 138 and 156 participants (for 15 or 20 predictor variables respectively) would be required. The greater of these figures was, therefore set for the target sample size to ensure that the calculation remained as flexible and conservative as possible.

By observing participation rates in similar studies (e.g. Lowe e ta l, 2003;

Osoweicki & Compass, 1998) it was possible to calculate an estimated response rate in the region of 40%. Accordingly, it was deemed necessary to approach in excess of 390 patients. Considering that 25% of patients were expected to be excluded from the study, the total number of diagnoses needed to achieve this sample size was estimated at approximately 520 new cancer diagnoses.

3£.2 Participant inclusion/exclusion criteria

All patients diagnosed with colorectal, breast, prostate or lung cancer from three North Wales NHS Hospitals (where clinical teams agreed to collaborate with recruitment) were considered for inclusion into the study.

Although any stage of diagnosis was included, any individuals perceived to have advanced disease or a very poor life expectancy at diagnosis (i.e. close to

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death) were excluded. Exclusion criteria also prevented the following individuals from being approached for the study:

• Those who had been diagnosed with cancer, but whom their

consultant hadn't yet informed, or who don't seem to understand the implications of their diagnosis.

• Those diagnosed with recurrent cancer.

• Those who had a poor understanding of the English language.

• Those concurrently suffering from any severe developmental, learning, or psychiatric conditions which may have impaired their understanding of the questionnaire.

• Any individuals unable to provide consent.

• Any individuals perceived to be at particular vulnerability or distress risk through participation (i.e. those perceived by the direct care team to have reacted to diagnosis with unusually high distress levels, to whom receiving this questionnaire may have a negative impact upon their ability to deal with their illness).

• Any individuals diagnosed more than six weeks prior.

Inclusion was determined by the CNS using a standardised inclusion matrix prior to referral into the study (appendix 3.24). This procedure was conducted in the presence of the researcher to minimise selection bias and a reason was

recorded for each individual excluded from the study. Reasons for patient exclusion are discussed in section 3.6.5.

3.6 PROCEDURE OF RECRUITMENT, DATA COLLECTION AND ANALYSIS