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Task
M-plus variable name Tformmaj which is the major scale of the transformational leadership. If significant relationship is detected between work group performance and transformational leadership’s major scale (Tformmaj), then work group performance (TPmajor) to be tested on each of the transformational leadership subscales ; Idealised influence (attributed) (Tformatt), Idealised influence (behavioural)(Tformbeh), Individualised consideration(Tformcon), Inspirational motivation (Tformmot), and Intellectual stimulation(Tformsti). The analysis should consider random slope model, and variance between random intercept and random slope.
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Transformational leadership is defined as a process that changes and transforms people, and comprises an exceptional form of influence, resulting in the achievement of higher levels of performance amongst followers than previously thought possible (Bass, 1990). People who exhibit transformational leadership often have a strong idealised influence (charisma), as well as a strong set of internal values and ideas. In addition, they are effective at motivating followers in ways that promote the greater good, as opposed to their own self-interest (Bass, 1990; Bass &Avolio, 1994). Bass andAvolio (2000) identify five components of transformational leadership traits and behaviours, which are theoretically and empirically related (Avolio& Bass, 1995). Those components are:
Work teams are increasingly becoming the primary means for organising work in contemporary organisations simply because teams are better able to provide a direct and collaborative effort to address complex tasks (Robbins & Judge, 2009). Generally, literature differentiates between two concepts: work teams and work groups. First, a work team is defined as a social group where its members are committed to a common purpose and working approach to which they hold themselves accountable (Guzzo&Shea, 1992). Working in a team generates positive synergies through coordinated efforts; thus, it is also defined as ‘a group whose individual efforts result in a performance which is greater than the sum of the individual inputs’ (Robbins & Judge, 2009, p. 323). On the other hand, a work group is defined as an aggregation of two or more people who are to some degree in dynamic interrelation with one another (McGrath, 1984). Work groups, as compared to work teams, have no need or opportunity to engage in collective work that requires collective effort, thus their performance is merely the summation of each group member’s individual contribution. There is no positive synergy that would create an overall level of performance greater than the sum of the inputs (Senior &Swailes, 2004). Moreover, the literature proposes several features that provide a foundation for a basic definition of work teams and groups. Scholars (e.g., McGrath, 1984; Kozlowski & Bell, 2013; Hackman, 1987) suggest that work teams and groups: (a) are composed of two or more individuals, (b) who exist to perform organisationally relevant tasks, (c) share one or more common goals, (d) exhibit task interdependencies, (e) interact socially, (f) maintain and manage boundaries, and (g) are embedded in an organisational context that sets boundaries, constrains the team, and influences exchanges with other units in the broader entity (Kozlowski & Bell, 2013).
Work teams and groups can also assume a wide variety of different tasks. McGrath (1984) suggests that group tasks could be distinguished over two dimensions. The first is the behavioural (action) tasks versus conceptual (intellectual) tasks. The second is the cooperation versus conflict tasks. Subsequently, he proposes four general group task processes. Firstly, the generate process that involves a group’s planning tasks such as generating plans and problem solving. Secondly, the choose process that includes group tasks which are either intellective or decision-making. Thirdly, the negotiate process that contains cognitive conflict tasks as well as mixed-motive tasks. Finally, the execute process that involves tasks which are either contests or performance tasks that involve striving to meet standards of excellence, with pay-offs tied to such standards rather than to victory over opponents. Furthermore, work group or team task performance concerns the degree to which a group or a team meets its goals and how well its output fulfils its mission (Hackman, 1987). It can be judged, in the most restricted sense, in terms of whether or not a team or a group achieves the task it has been assigned (Senior &Swailes, 2004). However, the organisations under this study, LPSOs, tend to rely on work groups in conducting their activities rather than on work teams (Rathboneet al, 2013).Consequently, majority of LPSOs work force are organised in work groups within organisational units. Those organisational units include departments, divisions, offices, and other smaller units (Annex A) which conduct a range of routine and bureaucratic activities that generally fall within the range of tasks outlined above by McGrath (1984).Therefore, it is critical for those organisations to understand whether their work groups’ performance is influenced by the group’s leadership style as well as by the characteristics of the members of the work group.
A quantitative survey strategy adopting cross-sectional design was used to collect data from leaders and followers within LPSOs. This strategy was selected owing to limited resources; a survey strategy offers a procedure for collecting data from a relatively large sample of leaders and followers at different levels and sites, rather than being restricted to qualitative analysis based on a small sample of employees. In addition, data collected using a survey might suggest relationships between variables that could lead to the production of models of such relationships. Surveys also enable researchers to have more control over the research process (Saunders, Lewis & Thornhill, 2007). The survey strategy is considered to be appropriate when variables cannot be manipulated (Bryman & Bell, 2007). Outcome variables (dependent variables), such as work group performance, cannot be manipulated easily and are therefore not ideal for experimental studies.
Although surveys are known to limit researchers to a set number of questions, constraining them to depend on others for further information, this method continues to be perceived in business and management research as the most economical and authoritative approach (Saunders et al., 2007). Hence, the author considered this design to be appropriate for exploring the research objectives of this study. The study used a structured survey format which drew upon existing validated instruments for the collection of data. It employed two different questionnaires that were distributed at the same time, and were to be completed simultaneously. One questionnaire asked leaders to rate their followers’ work group performance. The other asked followers to rate their leaders’ style.
This study adopts a non-probability sampling technique. Purposive sampling enables researchers to use their judgement to select cases that will best enable them to meet their research objectives. This technique is considered to be suitable when the researcher undertakes an in-depth investigation that focuses on a particular purpose (Saunders et al., 2007). Non-probability is common in the field of business and management studies, and is more prominent than probability sampling methods owing to the difficulty and costs involved with random sampling (Bryman & Bell, 2007). Non-probability sampling is also typically used owing to a study’s exploratory nature, limited access to the entire population, and time and cost constraints. Since this research focuses on investigating the relationship between leadership, followership and work outcomes – particularly in the Libyan public sector context – purposive sampling is perceived to be appropriate for the objectives of this enquiry.
Data was collected from leaders and followers from 24 LPSOs. In order to have an extensive representation of public sector institutions in the study, the surveyed organisations were chosen from a broad spectrum of major industries in the country to which the researcher had access. The organisations participated in this study were from seven key industries: finance, health, education, utilities, energy, manufacturing, and engineering and construction (see Table ?3?1).
Table ?3?1: Overview of the organisations and work groups that contributed to the study
No. | Sector | No. of Organisations | No. of Work Groups | No. of Replies | Types of Organisations |
1 | Finance | 6 | 22 | 95 | Banks, Investment, Insurance, Regulators |
2 | Health | 2 | 9 | 58 | Hospital, Health Centre |
3 | Education | 2 | 7 | 39 | University College, High School |
4 | Engineering and Construction | 5 | 29 | 142 | Consulting and Construction Firms |
5 | Utilities | 3 | 36 | 181 | Electricity, Water, and Environment Services |
6 | Energy | 3 | 22 | 87 | Oil, Gas and Renewable Energy |
7 | Manufacturing | 3 | 16 | 65 | Food, and Electrical Equipment Factories |
TOTAL | 24 | 141 | 667 |
All of the organisations in the study have a similar organisational structure: five to seven hierarchal levels composed of the chairman or CEO at level one; and deputy CEO, general managers, managers, heads of departments, heads of divisions, heads of offices, and heads of units at levels two to seven. Annex A provides a typical organisational structure of LPSOs. Data was collected from managers and employees of work groups across multiple organisational levels. The sample included 141work groups from various organisational units; each work group is composed of two or more individuals who perform administrative, technical or professional tasks. Due to the lack of measurable strategic goals in most of the surveyed organisations, the vast majority of those work groups do not have clear quantifiable objectives; consequently, they do not have reliable measures of their performance. Moreover, although some of the members in the work group share one or more common goals, they largely exhibit weak or no interdependency. Thus, this study relied on the judgement of each work group’s leader to evaluate the overall performance of his or her work group. Work group leaders included CEOs, general managers, mangers, and heads of divisions.
The initial sample contained data from 760 respondents out of 1000 questionnaires that were distributed, representing a response rate of 76%. At the analysis stage, 93 of the respondents were excluded from the analyses either due to severe damage to the survey papers, a missing work group identification code, or the entire questionnaire being unanswered. This resulted in 667 respondents (141 work group leaders and 526 followers) being retained (Table ?3?1). Responses with missing data were retained as the study examines the data using multiple-level modelling, which has been shown by Field (2009) as appropriate for handling missing data. The study also uses composition measurement for examining the relationship between the independent variables and work group performance. Thus, the missing data from some members of some groups in the sample (i.e., within-team none response) is important to consider and document because it can reduce external validity, statistical power (Newman, 2009) and cause potential bias of substantive hypothesis tests as well as a potential loss of research credibility (Maloney, Johnson, &Zellmer-Bruhn, 2010). However, the response rate to the work group performance items was 100% from all work group leaders (N=141). Thework groups in the sample included 2 to 12 respondents per group, with an average of 4.9 respondents per work group. There were full responses to group member survey (GMS) from more than 65% of the work groups’ members whereas the rest of the group members answered more than 52% of the survey.
The independent variable is transformational leadership, which is defined by five subscalesincluding; Idealised influence (attributed), Idealised influence (behavioural), Inspirational motivation, Intellectual stimulation, and Individualised consideration.
The dependent variable include a measure of work group performance adopted from Zellmer-Bruhn and Gibson (2006) that is used by the study to measure work group performance. There are seven demographic variables: gender, age, work experience, city, tenure in company, job role and education. This demographic data were collected in order to exclude their effect on the relationships amongst leadership, and followers’ workoutcomes.
This study used two surveys (paper and pencil approach) comprising existing and validated scales. The Group Leader Survey (GLS; Annexes A) was distributed to work group leaders, who were asked to rate their group’s performance. This survey included five questions drawn from the Team Task Performance scale (Zellmer-Bruhn & Gibson, 2006). The Group Member Survey (GMS; Annexes B) was distributed to work group members who were asked to rate their leaders’ leadership behaviours. This survey included 36 items of the MLQ-5X short rater form (Avolio& Bass, 1995). The researcher developed a hierarchal coding system to ensure respondent anonymity. Numerical codes identified work groups in each organisation. The same work group code was given to the GLSs and GMSs of members within the same work group. This procedure enabled the researcher to link leader and member answers in each work group. The following subsections briefly discuss the contents and validity of each of the six questionnaires used in this study.
3.4.1 Leadership
Leadership is measured using Avolioand Bass’ (1995) MLQ-5X short form. This instrument is one of the few measures available which assesses the Full-Range of leadership using a multifactorial model. Bass (1985) developed the original MLQ with the objective to measure both transactional and transformational leadership behaviours, and to accordingly investigate the nature of the relationship between these leadership styles and work unit effectiveness and satisfaction (Lowe &Kroeck, 1996). The MLQ-5X was introduced in 1991 and incorporated a variety of refinements (Avolio, Bass & Jung, 1999). The MLQ-5X is widely accepted as a valid and reliable tool, and has been used in more than 300 studies worldwide between 1995 and 2004 (Avolio& Bass, 2004). Based on a normative database created in 1999, the total-item reliabilities for each leadership factor ranged from ?=.74 to ?=.94 (Avolio& Bass, 2004).
The current study used the MLQ-5X short form (Avolio& Bass, 1995) to measure nine leadership components within the Full-Range of leadership model. The five components of transformational leadership were: idealised influence (attributed), idealised influence (behavioural), inspirational motivation, intellectual stimulation, and individualised consideration. Three components were transactional leadership: contingent reward (CR), management by exception active (MBEA), and management by exception passive (MBEP). One component was the laissez-faire leadership style. This study applied the MLQ-5X short rater form which contains 36 five-point items used by followers to describe their managers’ leadership skills (Avolio& Bass, 1995). Examples of items from the MLQ-5X short rater form include: ‘articulates a compelling vision of the future’ (transformational); ‘makes what one can expect to receive when performance goals are met clear’ (transactional); and ‘avoids making decisions’ (laissez-faire). The anchors used to evaluate MLQ-5X short factors are: 1 = not at all; 2 = once in a while; 3 = sometimes; 4 = fairly often; and 5 = frequently, if not always. MLQ-5X short scores were averaged for the items that comprise each scale: higher scores indicate greater perceptions of specific leadership behaviours (Avolio& Bass, 1995).
Researchers tested the nine-factor model across regions and established strong and consistent support for the Full-Range nine-factor model (Antonakis, Avolio&Sivasubramaniam, 2003; Avolio& Bass, 2004). In all cases, the nine-factor model produced the best fit and was consistent across both the region and rater. These findings provide a relatively sound foundation to examine a broader range of leadership styles – especially with regard to the MLQ-5X short factor structure – using a relatively large and diverse sample (Avolio& Bass, 2004).The researcher obtained the English and Arabic versions of the MLQ-5X short form from Mind Garden with their permission to use them in this study (Annex F). A specialist reviewed the Arabic translation to ensure its consistency with the Arabic spoken in Libya.
3.4.2 Work Group Performance
This dependent variable was measured using a five-item scale that captures group goal achievement and effectiveness (Zellmer-Bruhn & Gibson, 2006). The items are: ‘This work group achieves its goals’; ‘Thiswork group accomplishes its objectives’; ‘This work group meets its requirements’; ‘This work group fulfils its mission’; and ‘This work group serves its purpose’. The reported reliability of this scale is ? = 0.95. A principle component analysis (PCA) reveals that all items are loaded on a single factor with an eigenvalue of 4.23, which accounted for 84% of the variance. Factor loadings ranged from 0.88 to 0.94 (Zellmer-Bruhn & Gibson, 2006). Work group leaders rate their group’s task performance using a five-point Likert scale (1 = very inaccurate; 5 = very accurate).
The data used in this study includes replies from followers and leaders that are nested within work groups in two levels. First the lower level which is the individuallevel,also referred to as level-1 or micro-level. The data in this level includesan independentcontinuous variable; transformational, transactional (M-Plus variable name; Tformmajor).The other level of the data is the higher level which represents the work group leader’s level,also referred to as level-2 or macro-level. The data in this level includes leader’s demographics and one dependent continuous variable; work group performance(M-Plus variable name; TPmajor) which is assessed by the work group leader.
To account for the nested structure of the data, the study performed multilevel modelling analysis using M-Plus 7 software (Muthén&Muthén, 1998-2010). Prior to entering data to M-plus, data was first entered into SPSS version 23.0 for Windows where a missing value code (99) was assigned for all missing values of variables in the data set. Descriptive statistics were conducted to describe the sample demographics and the research variables used in the analysis. Frequencies and percentages were calculated for nominal data, such as gender, while means and standard deviations were calculated for continuous data, such as the subscales and major scales of leadership styles, and organisational commitment. Data was screened for outliers. The presence of outliers was tested by the examination of standardised value. Standardised values represent the number of standard deviations the value is from the mean. Values +/- 3.65 standard deviations from the mean were considered to be outliers and were removed from the data set (Pallant, 2013). Then, the data was exported into ASCII format that is acceptable for M-Plus. The next section details the multilevel modelling approach used to analyse the data.
Multilevel modelsare used to analyse clustered data, that is, data with a hierarchal structure with one response variable measured usually at the lowest level and predictor variables at all existing levels (Hayes, 2006; Field, 2009). A curtail problem in the statistical analysis of hierarchically structured data is the dependency of the observations from the same cluster at the lower levels since respondents from that cluster are subject to the same influence (Hox& Maas, 2005). If the hierarchal data was analysed on a single-level, the independence assumption underlying traditional statistical techniques (e.g., multiple regression, ordinary least squares regression and analysis of variance)is violated, which can affect parameter estimates and results in inaccurate statistical inferences (Hox, 2002).In contrast, multilevel models appropriately account for the hierarchal data structure that causing dependencies in the data and avoids standard error bias due to clustering that leads to inflated Type-1 error rates and incorrect confidence intervals (Hox& Maas, 2005; Field, 2009).In addition, multilevel models allow analysing variables at different levels as well as the analysis of cross-level interactions (Luk, 2004). Multilevel modelling also allow for sample size to be able to vary across levels which is a regular characteristic of nested data (Tabachnick&Fidell, 2006).
To examine the research hypothesisabove, the study employed multilevel regression analysisin two stages.First,null models are used to assess the appropriateness of the data for multilevel modelling and to establish a baseline for further model fit testing. Second,latent variable multilevel models(Croon &Veldhoven, 2007) are used to test the relationship between level-1 predictor (i.e., transformational leadership)and the level-2 outcome variable (work group performance) in the research hypothesis.
3.6.1 The Null Models
The first stage in the analysis usednull model (known also as intercept-only or unconditional model) to assess the variance in the dependent variables due to clustering, thereby evaluating the appropriateness of multilevel modelling for examining the data in the study.The residual variances from the null model are also used as a baseline for estimating the statistics importance of level-1 predictors in the multilevel models which is discussed in chapter 4. The general multilevel equation form for a null model for a level-1 work outcomevariable (Y) of follower (i) in work group (j) is given as follows (Snijders&Bosker, 2012):
Level 1 (within-group):Y_{ij}=?_{0j }+r_{ij}(3.1)
Level 2 (between-groups):?_{0j= }?_{00}+u_{0j}(3.2)
A mixed model formis derived by substituting equation 3.2 into equation 3.1:
Y_{ij}=?_{00}+u_{0j}+r_{ij}(3.3)
By substituting level-1 work outcome variablesinto equation (3.3), the following three null models (intercept-only models) are obtained and then examined in chapter 4:
Job satisfaction_{ij}= ?_{00}+u_{0j}+r_{ij}(3.4)
Organisational commitment_{ij}= ?_{00}+u_{0j}+r_{ij}(3.5)
Work engagement_{ij}=?_{00}+u_{0j}+r_{ij}(3.6)
Where the subscript (i) has the values 1,2,….n, the subscript (j) has the values 1,2,….,N. Also, n is the number of followers in the group j,Nisnumber of work groups in the study, ?_{0j} is the average work outcome for work group (j), r_{ij} is how a follower (i) in work group (j) differs from his/ her work group mean on work outcome, ?_{00} is the unweighted grand mean of work outcome across all work groups, and u_{0j} is the error term representing a unique effect associated with work group (j).
The null models in equations 3.4, 3.5 and 3.6above do not contain any predictor variables at all; hence the models do not explain any variance in the outcome variables.However, these models are important because they provide the basic partition of the variability in the data between the two levels (Snijders&Bosker, 2012). The quantity (?_{00+}u_{0j}) in the modelsis the random intercept containing a fixed component (?_{00}) and a level-2 random component (u_{0j}). The last component (r_{ij}) is the level-1 residual, also a random effect. The assumptions are that the level-2 random component (u_{0j}) is distributed normally with mean zero and variance ?^{2}_{u0}, and the residuals (r_{ij}) at level-1 are normally distributed with mean zero and variance ?^{2}_{r}inall groups, and both components are mutually independent.
The variance terms from the null modelsare used to compute an intra-class correlation coefficient ICC _{P1c }which is the ratio of the variance between the group level variance to the total variance. This type of ICC is also known as ICC(1)and is defined by the following equation (Shrout& Fleiss, 1979; Snijders&Bosker, 2012):
ICC _{P1c }= ?^{2}_{u0}/( ?^{2}_{u0}+ ?^{2}_{r}) (3.7)
The ICC value indicates the proportion of the variance explained by the grouping structure in the population; hence it informs whether the group is important in understanding the individual differences. An ICC of zero indicates that observations are independent of group membership. The larger the ICC, the more individual differences are due to differences between groups (Bliese& Halverson, 1998). In other words, a high ICC indicates that the between-group variance is larger than the within-group variance and the difference across responses is really from group differences, this supports the use of the multilevel analysis. In contrast, a low ICC indicates the variance is likely due to individual differences within a group (Bliese& Halverson, 1998), hence it does not support the use of multilevel regression. Hox (2002) suggests that ICC coefficients can be interpreted as follows: 0.05-0.09 indicates a low effect, 0.10-0.14 a moderate effect, and coefficients from 0.15 indicate a large effect.However, Mplussoftware automatically outputs the ICC as well as the estimated level-1 and level-2 variances for all variables included in the multilevel analysis.
3.6.2 Latent Variable Multilevel Models
The second stage of the analysis testsassociation between workgroup performance, whichis measured at level-2, and the predictor variable transformational leadership whichis measured at level-1.The traditional approaches to analyse this kind of relationship use a single-level analysis in one of the following two ways. The first method, involves aggregating level-1 predictors to the group level using the mean, then the level-2 outcomesare regressed on the aggregated level-1 predator. A major problem with this approach is that measurement error in the aggregated scores is not accounted for. Also, sampling fluctuation might be an issue when not all individuals within a group are investigated. The second method is a level-1 analysis performed in which the disaggregated outcome is regresses on the level-1 predictor. Disaggregation violates one of the basic assumptions of regression analysis, the independence between the units involved. Consequently, Type-1 errors are severely inflated leading to too liberal tests (Keith, 2006).
Understanding the limitations of the above approaches, the current study uses a latent variable multilevel model proposed by (Croon &Veldhoven, 2007) in which the model associates a latent variable with each of the predictor variables at level-1 and treats the individual scores on these variables as reflective indicators for that latent variable. According to Croon and Veldhoven (2007), a group level outcome variable (Y) for group (j) may be explained by one (or more) level-1 predictor (X) for individual (i) in work group (j) where a level-2 latent variable (?) is associated with the level-1 predictor (X) on which only the work groups have score (?_{j}). The relationship is given by the following multilevel models:
Y_{j}=?_{0}+?_{1}?_{j}+ r_{j}(3.8)
X_{ij}= ?_{j}+u_{ij}(3.9)
Substituting level-2 dependent variable work group performance into equation 3.8 and level-1 predictor transformational leadership into equation 3.9 the following latent-variable models are obtained for the research hypothesis:
Work group performance _{j}=?_{0}+?_{1}?_{j}+ r_{j}(3.10)
Transformational leadership _{ij}= ?_{j}+ u_{ij}(3.11)
In equation 3.9 above,the score (X) for each individual (i) in group (j) is treated as a reflective indicator for that unobserved group score. The intercept is (?_{0}) and the slope is (?_{1}). The error term(r_{j}) is assumed to be homoscedastic, that is, to have a constant variance ?^{2}_{r} for all work groups. The variance of the unobserved latent group score (?) is denoted by (?^{2}_{?}), the variance of the disturbance term (u_{ij}), assumed to be constant for all subjects and groups, by (?^{2}_{u}). Moreover, (r_{j})and (U_{ij})are assumed to be mutually independent and to be independent from the group variable (?). Prerequisites for carrying out an analysis according to this model are that the number of groups is not too small and that ICCs for the level-1 variables are sufficiently large (Croon &Veldhoven, 2007).
3.7 Estimation of Regression Parameters
The estimation of statistical parameters, regression coefficients and variance components, in multilevel modelling is generally done using maximum likelihood (ML) method (Hox& Maas, 2005). There are two different varieties of maximum likelihood estimation are commonly used in multilevel regression analysis. The Full Maximum Likelihood (FML) where both the regression coefficients and the variance components are included in the likelihood function. The other is Restricted Maximum Likelihood (RML) where only the variance components are included in the likelihood function. The difference between the two is that FML treats the estimates for the regression coefficients as known quantities when the variance components are estimated, while RML treats them as estimates that carry some amount of uncertainty (Goldstein, 1995; Bryk&Raudenbush, 1992). Since RML is more realistic, it should in theory, lead to better estimates, especially when number of groups is small (Bryk&Raudenbush, 1992).
The maximum likelihood procedure generates standard errors for most of the parameter estimates. These standard errors can be used in significance testing, by computing the test statistics Z:Z= parameter/(standard error parameter). This statistic is referred to the standard normal distribution to establish a p-value for the null-model (Hox& Maas, 2005). The Maximum likelihood procedurealso producesa statisticscalled the deviance (the deviance equals -2 times the log-likelihood), which indicates how well the model fits the data. In general, models with a lower deviance fit better than models with a higher deviance. In addition to the standard errors, the deviance can also be used to test parameters for significance. If two models are nested meaning that a specific model can be derived from a more general model by removing parts of from that general model, the deviances of the two models can be used to compare their fit statically. For nested models, the difference in deviance has a chi-square distribution with degrees of freedom equal to the difference in the number of predictors that are in the two models. The deviance test can be used to perform a formal chi-square test, in order to test whether the more general model fits significantly better than the simpler model. The chi-square test of the deviance can also be used to good effect to explore the importance of a set of random effects, by comparing a model that contains these effects against a model that excludes them (Hox, 2010).
The default choice of estimator for multilevel analysis in M-Plus is Robust Maximum Likelihood (MLR), which is preferred when continuous outcome variables are not clearly normally distributed. Likert-type categories, as the ones used by the current study, are typically best treated using (MLR) since this estimator adjusts the important inferential elements of the results; hence it is used to estimate the parameters for the studied models.In addition, M-Plus uses full information maximum likelihood (FIML) estimation which includes the missing data points in the analysis hence there is no need to remove subjects with incomplete subject data(Muthen&Asparouhov, 2002).
3.8 Sample Size and Power
In addition to the estimation method, the sample size might affect the accuracy of the estimates of the multilevel model. The maximum likelihood described earlier is asymptotic which translates to the assumption that the sample size must be sufficiently large (Maas &Hox, 2005). In multilevel regression, however, there is a sample size for each level, defined as the total number of units observed for this level. For testing the effect of level-1 variable, this level’s sample is of main importance, similarly if we test the effect of level-2 variables it is this level’s sample size is of main importance. The average cluster sizes are not very important for the power of such tests (Snijders, 2005). This implies that the sample size at the highest level is the main limiting characteristics of the design. There are, though, two sample size issues to be concerned about. One issue has to do with the minimum number of cases needed to for using multilevel regression to avoid biases. The other issue concerns sufficient statistical power needed for obtaining significance. According to Maas andHox (2005), a minimum of 30 cases at the group level of analysis is needed for adequate power in multilevel modelling when considering contextual effects, while at least 50 cases are needed for the correct estimates of standard errors. Following this rule, the current study has a sufficient amount of cases at the work group level (N=141) required for robust estimations.
3.9 Centring of Independent Variables
Another important consideration when using multilevel regression analysis is the centring of the independent variables. Centring refers to the process of transforming a variable into deviations around a fixed point. There are two forms of centring that are typically used in multilevel modelling; group mean centring and grand mean centring. Group mean centring means that for a given variable we take the score and subtract from it the mean of the scores of that variable within a given group,while in the grand mean centring we subtract from variable’s score the mean of all scores (Field, 2009). In this study predictor variables are grand mean centred,where for each independent variable X the mean of the scores is subtracted from the raw score X_{ij}to produce centred scoreX^{C}_{ij}.In the analysis, the centred scoresX^{C}_{ij} are then used instead of the raw scores. By centring the predictor variables before the analysis the zero point of these variables become more meaningful, especially, when predictors do not have a meaning zero point. Moreover, multilevel models with centred predictors tend to be more stable, and estimates from these models can be treated as more or less independent from each other (Field, 2009).
4.1 Model fit
4.2 The strength of the relationship
4.3 The association between the random slope and random intercept
A gathering level variable (Y) is clarified by level-1 indicator (X) for individual (i) in work bunch (j). Here on level-2 inert variable (?) the work bunches have score (?j). The relationship is:
Yj=?0+?1?j+ rj (4.1)
Xij= ?j+uij (4.2)
By substituting subordinate variable in condition 4.1 and indicator transformational administration in condition 4.2, following models are obtained:
Work bunch execution j=?0+?1?j+ rj(4.3)
Transformational initiative ij= ?j+ uij(4.4)
For condition 4.2,the score for every person is dealt like an intelligent pointer in secret gathering score.
Here, ?0 stands for capture and ?1 stands for slant. The ‘rj’ is blunder term, which is homoscedastic. The notation (?2?) indicates change of in secret idle gathering score, whereas, the notion (?2u) indicates fluctuation in unsettling influence term. Both (rj)and (Uij)are autonomous and free of the gathering variable. Essentials of completing an examination include appropriate quantity of gatherings and ICCs.
4.4The variation among the random slopes and intercepts
yij = ?0 + (?1 + u1ij)x1ij + u0j + e0ij
By rearrangement:
yij = ?0 + ?1 x1ij + u0j + u1j x1ij + e0ij u0j u1j
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