Introduction he current educational system in Ethiopia is organized in cycles or levels of formal schooling that includes ten years of general education. General education is completed at the end of the first cycle of general secondary school education (Grade 9 and 10). Moreover, this cycle is intended to enable students to identify area of interest (Natural Sciences Stream and Social Sciences Stream) for further training in the second cycle of general secondary education (Grades 11and 12) to prepare students for continuing their studies at higher education level (University or collages) or selecting their own vocations. Students appear for the New National Examination at the end of grade 10 (normally under 18 years old) which is known us the Ethiopian General Secondary Education of Certificate Examination (EGSECE). This is after the students have successfully achieved school examinations in all school subjects. However, students should score a minimum of 2.00 on a scale of 4.00 in EGSECE or a minimum of 50 out of 100 in standard school exams at least in five or seven subjects: English, Mathematics (both compulsory) and any other three or five science (Natural or Social) subjects in order to appear in EGSECE [14]. Girls' education is one of the fundamental pillars for ensuring sustainable economic development, democratic participation and poverty reduction. As a result, gender discrimination affects not only women but also the overall growth of the economy. In this connection, the Ethiopian government has given more attention to girl's education. In 2003In -2004, due to the favorable policy environment, the gross enrollment of female students at general secondary first cycle (9-10) was about 37.0% and at the preparatory level (11)(12), it was 29.0%. Moreover, in technical, vocational and training institutions (colleges), it was 49.0%, whereas it was 25.2% in higher education/Universities. Nevertheless, there was a great variation of students' achievement at different school type (non-governmental and governmental) based on their gender. Without controlling for student background differences, nongovernmental schools scored higher than government (public) schools ( [4]; [13]). It is obvious that students at schools can be classified as clever (high achievers), medium (average achievers) and lazy (low achievers) with respect to individual's achievements in specific school subjects based on exam scores or general test results of subjects. The general belief is that, if the student is intelligent or clever, he/she is expected to perform well at school in compulsory and science school subjects and is well fitted for national and regional exams. But intelligence is not the only influential factor of academic achievement in school subjects. In addition to intelligence, there are various factors influencing academic achievement of students at school in each school subjects ( [2]; [19]). This study has been undertaken to investigate multivariate evaluation of the impacts of family with student and school characteristics variables on academic achievement of students on five selected subjects at secondary schools, specifically in grade 10. The presence of all or some of the factors identified above may have resulted in the poor academic achievement of students on each school subjects in some areas of our country. However, evidence of the availability of these factors as well as other factors need to be obtained or checked. The purpose of this study, therefore, is to obtain the factors that are responsible for the poor academic achievement of students with school type and gender gap on school subjects among secondary schools of grade 10 students at Hawassa city, in SNNPR state. # Statement of the Problem In 2007/08 the number of students who sat for grade 10 national exams, at SNNPR state, was 92,836 (male 61,742 and female 31,094). Out of these who get CGPA of 2.00 and above out of 4.00 were 33,211 (25,085 males and 8,126 females). The percent of promoted students in a successive three years, 2005/06, 2006/07 and 2007/08, were 45.8%, 44.2% and 35.8%. Specifically, the percentage of promoters (scored 2.00 and above) at Hawassa City Administration in 2007/08 were 46.7 %. [12] Reported that the test items (exam questions) of the EGSECE for English were not relatively content valid. Hence, test items did not match with the syllabus contents. Students might pass from one class level to the other as they evaluated on CGPA result of all subjects. But due to achievement variation with respect to each school subject, students get difficulty and being unsuccessful in higher level education which leads directionless. The current education system of Ethiopia gives a great attention, about 70%, on natural sciences subjects, to enhance sciences and technology. Therefore, it is better to find solutions to the problems and factors one faced in his/her academic achievements in selected subjects: Mathematics, Biology, Physics, Chemistry and English at secondary schools in grade 10 distinctly but dependably. Many reasons have been attributed for the high failure rate and poor academic achievements in secondary schools. Some researchers traced that the high failure rate of students was due to student's inability to comprehend and balanced the principles of some subjects such as Mathematics, Physics and others. Others are of the view that the abysmal school achievement is due to loaded curriculum (there is too much to be taught within a short time) ( [8]; [12]). Again some people suggest it on lack of proper supervision on the part of school administration and family control in student's self-carelessness ( [7]; [10]). Likewise, [13] claimed that gender stereotype and student's interest to the subjects have also great influential effect. Peculiar nature of some factors and the students low and unbalanced success rate have led to this study on the multivariate analysis of the determinants of students' academic achievement measured in five selected subjects at general secondary school completion level, first cycle, grade 10. The following research questions have been developed to guide this study: # b) Objectives of the Study The general objective of the study has been to determine the key factors influencing academic achievements of students measured in exam scores of five subjects in grade 10 (Mathematics, Biology, Physics, Chemistry and English), and to assess the variations accounted at school and individual (student) level for each response (school subjects). The Specific Objectives are ? To identify the most important factors (covariates) influencing academic achievements of student's in each component of selected subjects in grade 10. ? To determine the relationship among the school subjects at both school and student level; and whether there is gender and school type differences in this relationship. ? To quantify and determine the within and between schools variation for each components of selected ? To determine the groups or clusters of interrelated observed variables or items as component factors that explain the variation of achievement indictor variables. # II. # Materials and Methods # a) Description of the Study Area and Population The study was conducted in Hawassa, the capital city for SNNPR state, which was established in 1960. It is located at about 275 km South West of Addis Ababa, and near to Hawassa Lake. Geographically it lies between 07 0 05 ' Latitude North and 38 0 29 ' Longitude East. According to the report of [6], the estimated population size of the city (urban) in 2007 was 159,013 out of which 81,984 were males and 77,029 females. There are 4 governmental colleges and one university, 8 non-government (private) colleges, 5 governmental high (secondary) schools, about 15 nongovernmental high (secondary) schools. The gross enrollment rate of secondary school students at Hawassa Town Administration has been 62.1%. The target population for this study was grade 10 students of both government and non-government schools registered in 2010-2011 academic year at Hawassa City secondary schools. The total population of students in all high schools of the city was 6,384 in 2010-2011 academic year. Exclusion criteria were made on the students who were transferred to other schools or those dropped out, only completed enrolment procedures at the school but did not yet attend the national exam or left the school or had been absent for more than four continuous weeks (excluding school vacations) and had no examination results in 2 of the most targeted school subjects (compulsory subjects). This was because full information about those students was not available. # b) Sampling Design and Procedure A cross-sectional study with stratification sampling designed to take independent samples for different sub-populations was conducted. The stratums were governmental and non-governmental secondary schools as school type. Sampling methods are scientific procedures of selecting those sampling units which would provide the required estimator with associated margins of uncertainty arising from examining only a part not the whole of the population. The main purpose of stratification is to reduce sampling error. Moreover, stratified sampling is a technique which uses any relevant information that might be available in order to increase efficiency. It involves the division or stratification of a population by partitioning the sampling frame in to non-overlapping and relatively homogeneous groups [5]. A list of grade 10 students was obtained from Hawassa City Administration Education and Capacity Building Department. The population of grade 10 students was stratified into governmental and nongovernmental school and the required sample size for the study was determined from each stratum. The multistage sampling procedure was employed as: The selection of a simple random sample was usually carried out according to a set of mechanical instructions which guarantees the random nature of the selection procedure. This is an equal probability of selecting individual units for all elements in the population of the school. Stage three: simple random sampling of students from class Taking a list of students with their registration number in each school, then refer to a table of random numbers; the required sample students were selected. In simple random sampling, the selection of one individual was independent of the selection of another individual. i # . Sample Size Determination In the planning of a sample survey or researches, a stage at which a decision must be made about the size of the sample is always required. However, too large a sample implies wastage of resources, and too small a sample diminishes the utility of the results. Therefore the decision should be made with a minimum cost but the estimate will explain the population characteristics with a high probability. However, several formulas developed for sample size calculations that conform to different research situations [5]. The sample size for this study was determined based on stratified sampling with proportional allocation at 95% confidence level using the general formula for sample size determination adopted as: The known methods of estimating for calculating sample size of any survey were by taking the sample in two steps; one by the results of a pilot survey and another by previous studies sampling of the same or similar population and guesswork about the structure of the population [5]. But for the present study, and the margin of (absolute) error were determined from the results of previous studies of similar population. The sample variance 2 1 s = 0.20885 and mean = 2.62 were taken for government school from the study which assessed the determinants of students' academic performance in government schools of grade 10 at Hawassa town taking a sample of 920 students (Hanna;. Then, for this study was calculated as: = On the other hand, the sample variance 2 2 s = 0.13421 was taken for non-government schools from the previous study at the same area [11]. The total population was (number of students in 11 selected secondary schools of grade 10) 5006 = N from 5 governmental and 6 non-governmental selected secondary schools, which contained total number of grade 10 students in governmental schools # c) Methods of Data Collection In assessing the academic achievement of students' measured by exam results scored in school subjects, Mathematics, Biology, Physics, Chemistry and English at both government and non-government sample secondary schools, both primary and secondary data were used. The primary data was collected using questionnaire method. The questionnaire consisted the student's, family background and school characteristic variables on the student's academic achievements evaluated in selected 5 subjects. Individuals sampled for this study were asked to complete the determinants of students' outcome (in five school subjects) study questionnaire. The secondary data on academic achievements of respondents was measured by their ? ? = = + = 2 1 2 2 1 2 2 1 h h h h h h h S W N V W S W n N N W h h = h n 2 2 ? ? ? ? ? ? ? ? ? ? = ? V ? ? ? ? ? ? ? ? str Var h h h str y W ? = ? ? = ? 2 1 h n i hi h n y y h ? = ? = 1 = = ? ? = ? h N i hi h N y h 1 ( ) 2 1 2 1 ? ? = ? = h N i h hi h N Y y S h 1 y 2 y 96 . 1 025 . 0 2 = = z z ? 2 s 2 s ? 1 y ? n S 2 2 ? ? = ? 0295 . 920 208849 . 0 96 . 1 = ? ? ? ? ? ? ? ? N N W h h = 2 h S n 0 n ? = ? ? ? ? ? ? ? ? ? ? 2 1 2 2 2 h h h S W ? N n n n 0 0 1 + = n N N n h h ? ? ? ? ? ? = 2 , 1 = h Year 2016 where h = stratum the EGSECE results (scores) in each of the five selected subject (Mathematics, Biology, Physics, Chemistry and English). Besides, school records with regard to students' exam registration number and some profiles of teachers and schools were taken from record offices of the schools. Sampled grade 10 students were taken with their exam scores of all five school subjects and the student's results were standardized and scaled to be 4.00. # d) Variables of Interest in the Research The outcome variables used in this study were the five selected school subjects as individual's achievement measures using EGSCEE results or scores on the five school subjects (Mathematics, Biology, Physics, Chemistry and English). All achievement scores were taken as standardized and transformed to assure that all scores were scaled in the same metric. This also allowed us to interpret the between school variances as the percentage of variation in student achievement accounted for by schools in PCFA, MVML and multivariate multiple linear regression analysis with respect to each response. The set of explanatory variables included were the composite common factors of students, family, teachers and schools characteristic variables. i. Students and Family Characteristic Variables These were: Age, gender, religion of student, parents' employment status, natural talent, students' job aspiration, time spent on study, peer(group) effect, student class attendance(absence), skipped class, student's satisfaction with school administration, satisfaction with school rules and regulations, academic confidence, preferred study time, preferred study place, distance of the school from students' home, availability of text and reference books at home, home location, parental involvement, fathers'/guardians' level of education, comfort of study place at home, mothers' education level, average family expenditure, other expenses related to education, satisfactions in food type available in home, pervious grade scores, students attitude and perception on school subjects (difficulty, boringness, preference, etc.). ii. School Characteristic Variables These were: teachers average workload, average year of experience, teachers average educational level, teacher preparation, class size, teaching method, standard of examination, parent to teacher communication, teacher absence, teacher late, average size of school, school fee, completion of the syllabus, school type, student-teacher ratio, teacher efficient and skills, school location/environment, current curriculum, human resources (teachers per subjects, principals, supervisors), infrastructure (buildings, classrooms, sport facilities), library facility, equipment (desks, blackboard, telephone, duplicating computers), amentias (toilets, electricity, water), and availability instructional materials (text and reference books, maps and charts), laboratory facilities, academic counseling service, health service (first aids). # III. # Methods of Data Analysis a) Factor Analysis Model This analysis describes the covariance relationships among many variables (items) in terms of a few underlying and unobservable random quantities. The observable random vector X with P components has mean ? and covariance ?. The factor model postulates that X is linearly dependent upon a few unobservable random variables The factor analysis model is given by: X = LF+ ?, where is a matrix of unknown constants called factor loadings. # L pxm = F= and = The coefficient is the loading of the variable on the factor. i. Assumptions of Factor Model 1. E (F) = 0 = (0, 0, ?,0 ) T 2. cov (F)= E (FF T )=I m 3. E (?) = 0= (0, 0,?, 0) 4. Cov ( ? )= E(? ? T )= ? pxp , ? is a diagonal matrix 5. Cov (? ,F) = E(? ,F T ) = 0= (0,0,?,0) T ii. Covariance Structure for Orthogonal Factor Model m p PxM L ? ij l th i th j ? .... ????. ? 11 l 12 l m l 1 21 l 22 l m l 2 1 p l 2 p l pm l ? 2 f 1 f m f p ? ? ? ..... 2 1 Year 2016 1. Cov(X) =LL T + 2. Var = , where is the specific factor. ? ( ) i X i im i i l l l ? + + + + 2 2 2 2 1 ... i ? th i 3. 4. ( ) ij j i l F X Cov = , 5. , loading matrix. Communality is defined by: The factor model assumes that variables and covariance for X can be reproduced from pm factor loadings and p specific variables . The factor model provides a simple explanation of the covariation in X with parameters which are fewer than parameters in ?. # iii. Methods of Estimation of Loading If the off diagonal elements of sample covariance S are small or those of the sample correlation matrix R essentially zero (identity matrix), the variables are not related. This implies that a factor analysis will not prove useful and in these circumstances, the specific factor plays a dominant role. If covariance matrix appears to deviate significantly from a diagonal matrix, then a factor model can be entertained and the initial problem is one of estimating the factor loading and specific variance . There are two popular methods of parameter estimation, Maximum Likelihood (ML) Method and Principal Component Method. However, for this study, the principal component method was used. # iv. The Principal Component Method The spectral decomposition of covariance ? having eigenvalues-eigenvector pairs with is given as . From the above equation, we can obtain the loading, . I T T TT = = was a matrix of 'rotated' loadings, where is the identity matrix. This shows that the estimated covariance (correlations) matrix remains unchanged since ? ? ? ? ? ? ? ? ? ? + = ? + = ? + * ' * ' ' ' L L L TT L L L . A useful byproduct of factor analysis was factor scores. Factor scores were composite measures that can be computed for each individual on each common ( ) km im k i k i k i l l l l l l X X E + + + = ... , 2 2 1 1 ( ) L F X Cov = , 2 2 2 2 1 2 ... im i i i l l l h + + + = ( ) ( ) 2 1 2 1 + = ? + p p p p p ij l i ? ( ) pm p + ( ) 2 1 + p p ij l i ? ( ) i i e , ? 0 ...+ + + = ? ... 2 2 2 1 1 1 = L ( ) S tr S S S pp = + + + ... 22 11 ? ? ? ? = + + + ? ? ? p ... 2 1 ? i ? p i ,....., 2 , 1 = 0 ... 2 1 ? ? ? ? ? ? ? m ? ? ? ? L pxmpi p i i i ik x l x l x l x l f ? ? ? ? ? + + + + = ... 3 3 2 2 1 1 , where # b) Multivariate Multiple Linear Regression Model The multivariate extension of multiple linear regression was used to model the relationship between responses and a single set of predictor variables . Each of the response was assumed to follow its own regression model, so that ? i r ri i i i i i i z z z ? ? ? ? ? + + + + + = ? ...( ) = ? ×m n = ( ) ( ) m r × +1 ? = = ( ) m n× ? = = m M Y Y Y ,..., , 2 1 r r z z z ,..., , 2 1 m 0 .... 2 1 = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? m E ? ? ? ( ) ? = ? Var th j 10 ? 11 ? ? r 1 ? 20 ? 21 ? ? r 2 ? ???? 0 n ? 1 n ? ? nr ? 11 ? 12 ? ? m 1 ? 21 ? ? m 2 ? ??? 1 ? 2 ? ? ? 22 ? ( ) | 1 ? ( ) | 2 ? ? ( ) m | ? 01 ? 02 ? ? m 0 ? 11 ? 12 ? ? m 1 ? ??? 1 r ? 2 r ? ? rm ? ( ) ( ) | | 2 1 ? ? ?| ( ) m ? 11 ? 12 ? ? m 1 ? 21 ? 22 ? ? m 2 ? ?? 1 n ? 2 n ? ? nm ? ( ) | 1 ? ( ) | 2 ? ? ( ) m ? | The multivariate linear regression model is: with 0 = ? ? ? ? ? ? ? ? and ( ) ? × = ik Cov ? ? ? k i , for . ,..., 2 , 1 , m k i = The 'm' observed responses on the j th trial (student) have covariance matrix ( ) i ' 1 ' i ? ? ? ? ? ? ? ? ? ? = ? ? ? . Then collecting the uni-variate least squares estimates yields: ? ? = ' ' ? ? ? ? ? ? ? ? ? ?1 = ? ? ? ? ? ? ? ? ? ? ? ? ? ' 1 ' . Using a matrix ? ?, one can easily ascertain that the matrices of predicted values: ? ? ? ? ? ? ? ? ? ? ? = ? = ? ? ? ? ' '1 ? and residuals: . ' ' ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? = ? ? ? 1 ? If the model is of full rank, rank (Z) = r + 1 < n; and ? and ? are also uncorrelated. Furthermore, because ? ? ? + = ? ? , then one have ? ? ' = ? ? ? ? ' + ? ? ? ? ' Residual SSCP= ? ? ? ? ? ? ? ? ? ? ? = ? ? ' ' ' and the unbiased estimator of ? is 1 r n ' ? ? ? ? = ? ? ? ? . ii. # Test of Hypothesis The hypotheses of all explanatory have no effect on academic achievements of students jointly on the responses, i.e. the th i school subject doesn't depend on the 'r' explanatory variables: ( ) ( ) ? . S t i , s i , s cal ? ? ? ? ? ? ? ? = ? ? , where ( ) ( ) ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? i s i s E S , , # var . , . Decision Rule: if or p-value less than 05 . 0 = ? , we reject the null hypothesis. On the other hand, the confidence ellipsoid for ? can be easily contracted with the one-at-a-time t value ( ) 2 1 ? ? ?r n t and using intervals ( ) ? ? ? ? ? ? ? × ? ± ? ? ? ? ? i 1 r n i SE 2 t . Here if ( ) ( ) ( ) ( ) ( ) ( ) nxm xm r r nx nxm ? ? + ? = ? + + 1 1 k i, ? k i, ? B m i ,..., 3 , 2 , 1 = ( ) ( ) 1 + ? r n t ( ) ? ? ? ? ? ? ? i Var ? ( ) 1 ' , ? ? ? ? = i i ? ( ) ( ) ( ) ( ) ? ? ? ? ? ? = ? ? ? ? i r i i i Var Var Var Var Diag , 2 1 0 ,..., , , ? ? ? ? ( ) ( ) 1 ' , ? ? = ? ? ? r n i i i i ? ? ? ( ) ( ) # iii. Checking the Goodness of Fit of the Model It is imperative to examine the adequacy of the model before the estimated function becomes a permanent part of the decision making apparatus [9]. All the sample information on lack of fit is contained in the residuals. . # iv. Residuals The residuals are defined as: ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? = ? ? ? ? ' 1 ' Since a residual may be viewed as the deviation between the data and the fit, it is also a measure of the variability in the response variable not explained by the regression model. Plotting residuals is a very effective way to investigate how well the regression model fits the data and to check the assumptions. v # . Normal Probability Plot The most commonly used methods of checking normality of an individual variable are the Quantile-Quantile plot (Q-Q plot), P-P plot and Normal Curve Histogram. The P-P plotted as expected cumulated probability against observed cumulated probability of standardized residuals -line should be at 45 degrees. The variable is normality distributed if this plot illustrates a linear relationship. In case of the assumption that says the combinations of variables follow a multivariate normal distribution, one can generally test each variable individually and assume that they are multivariate normal if they are individually normal [3]; [1]). vi. Ethical Issue/ Considerations Ethical approval was obtained from research ethics committee of Hawassa University, Postgraduate school of Computional sciences. Following the endorsement by the research ethics committee and acceptance of the postgraduate school and statistics department, Hawassa City Administration Education and Capacity Building Department was informed about the study through a support letter from Hawassa University research Postgraduate research office. Then verbal permission had been obtained from respective department of the city administration. Following the endorsement by Hawassa City Administration Education and Capacity Building Department, the selected schools were informed about the objective of the study through a support letter from Hawassa City Administration Education and Capacity Building Department and oral permission and supports were obtained from the respected school principals, teachers and students. As the study was conducted through review of academic records, the individual person was not subjected to any harm as far as the confidentiality is kept. Consent was obtained from individual person or student who was selected to fill the study questionnaire. To preserve the confidentiality, data recorders or file keepers, in the City Administration Education and Capacity Building Department extracted the data from the academic records. Moreover, no personal identifiers were used on data collection form. The recorded data was never accessed by a third person except the principal investigator, and was kept with a firm confidentiality in a secured place. IV. # Results # a) Descriptive Results From the results in Table 3.1, the average academic achievements of students measured in Mathematics, Biology, Physics, Chemistry and English subjects for non-government school students were, respectively, 2.99, 2.97, 2.50, 2.88, and 3.14 with standard deviations 0.822, 0.899, 0.942, 0.806 and 0.805, respectively, and that of government schools were 2.61, 2.73, 2.24, 2.74 and 2.77 with standard deviations 0.838, 0.866, 0.964, 0.872 and 0.802, respectively. Table 3.2 shows the mean academic achievements and the coefficient of variations for the five subjects. In terms of coefficient of variation, the variability was the lowest for English and highest for Physics subjects. This may indicate that students' achievements were most consistent for the English subject and least consistent for Physics subject. Physics was considered as difficult subject for many students. # b) Results of Factor Analysis Before conducting the central MVML and multivariate multiple regression analyses it is important first to establish the psychometric properties of the instrument used. Principal Component Factor Analysis was done in two steps. The first one was a general PCFA that considered the socioeconomic and demographic variables with general school characteristic variables and the second was a separate PCFA relative to each achievement measures of the five subjects. This provided component factors for each of the five school subjects each based on the subject related observed items as students' responses on their personal, school and teacher characteristic variables relative to school subjects. The overall reliability was computed to be Cronbach's alpha=0.724 indicating that the questionnaire items were consistent. The KMO statistic values test if sufficient items (by partial correlation among variables) are available for each factor component in the factor analysis. KMO statistic for the separate PCFA with respect to the school subjects Mathematics, Biology, Physics, Chemistry and English were 0.81, 0.77, 0.82, 0.78 and 0.84, respectively; with the general PCFA of 0.79. These were all greater than 0.5 indicating that the sampling was adequate for factor analysis and there were significant relationships among the perceived factors of achievements in the school subjects. The data were also checked for Bartlett's test of Sphericity to see that the original variables were sufficiently (bi-variate) correlated and these met the criteria with 0 . 11170 These indicated that the original observed variables were sufficiently correlated (the variables were not completely uncorrelated) and factor analysis was possibly appropriate in each case. The output matrixes contained the loading of each variable onto each factor. All loadings less than 0.5 were suppressed in the output and so were blank spaces for many of the loadings. Thus, the loadings were acceptable and easy for interpretation. The results of separate factor analysis (with factor loadings greater than 0.5) are presented in Tables 4, 5, 6, 7 and 8 of Appendix-1 and Figures 1 in Appendix-2 of the Scree plots. The criteria that the required amount of explained variation accounted for being large, logical interpretability of factors and Scree plot tests were considered with Kaiser Criteria. Kaiser criteria is accurate when there are less than 30 variables with lager sample and communalities after extraction being greater than 0.6. Depending on the correlation matrix and communalities, some observed variables were rejected. Of all 140 observed items, using principal component extraction and Varimax rotation, the study found factor solution of the 28-variables for each subject. Then, six underlying common factors were obtained for each separate factor analysis of Biology, Physics, Chemistry and English related items that constituted or explained 76.67%, 78.80%, 68.64% and 73.43% of the total variability in the corresponding original observed variables, respectively. There were four common factors for Mathematics related items which constituted or explained 77.38% of the total variability in the original observed variables related to Mathematics. Factor scores of each component factor for each of the 719 individual respondents were computed and these scores were used as data for further analysis. The common factors obtained from the general and separate PCFAs which were used as covariates, ? School facilities (SF), Interest (InterstS) to the subjects, ? Family status (FS), Motivation (MotivS) to the subjects, ? School volume (SV), Trouble (TroubS) to the subjects and ? Safe reading (SafR), Self-concept (SelfC) to the subjects. ? Loadings Less than 0.5 were suppressed. # c) Results of Multivariate Multiple Linear Regression Analysis Multivariate multiple linear regression analysis was used to examine the effect of independent variables or factors on the outcome variables, i.e. academic achievement in selected subjects. Most of the explanatory variables were the common factors obtained from the general PCFA and some were the regularly appeared component factors in each separate PCFA. The results are shown in Table 3.6. In this analysis the overall determinants of academic achievement were assessed in terms of the five school subjects to identify the basic determinant factors for both government and non-government schools taken together. The factors sex, school type and school facilities (SF) were found to be jointly statistically significant for achievements in all the five selected school subjects. Family status (FS) was significant for achievements in the four school subjects (Biology, Physics, Chemistry and English) but statistically insignificant for achievement in Mathematics. School volume (SV) has a significant influence on achievements in the two school subjects Biology and Chemistry. Interest to the subjects (InterstS) has a significant influence on academic achievements of students in Biology and Physics. Moreover, the factors trouble to the subject (TroubS) and motivation (MotivS) to the subject in terms of Mathematics, self-concept (SelfC) in terms of Physics and students future aspiration (FutureAspira) in terms of Physics and Chemistry had significant impact on student's academic achievement as observed in overall combined data of government and nongovernment schools. Moreover, the factors such as sex, interest to the subject (InterstS), motivation to the subject (MotivS), self-concept (SelfC), family status (FS), school facilities (SF) and future aspiration (Future Aspira) had positive impacts on students' academic achievements of the school subjects. However, trouble (TroubS) of the subjects and school volume (SV) showed significant negative impact on students' achievements of all the five subjects. # Volume XVI Issue # Discussions and Conclusions The PCFA technique was used as separate PCFA of items with respect to the each five responses and the general PCFA incorporated other general student with family and school with teacher characteristics variables in the data reduction. The multivariate single level multiple linear regression was applied on overall schools data. The results obtained are discussed as follows: On an average, students, in non-government secondary schools, performed better than those in government secondary schools in almost all the achievement measures of the five school subjects. This might be because of higher availability of school and home educational supply and facilities, better study positions and higher parental involvement with teachers and students at the schools as compared to that at government schools. Moreover, on overall average, male students achieved better in almost all school subjects than female students. This implied that the school and family might treat gender differently and the variation in students' personal factors such as trouble to the subjects, self-concept, interest and motivation to the subjects showed significant impact on students' achievement ( [4]; [6]; [20]). The results obtained from the separate PCFA in each achievement measuring response indicated that about four factors related to Mathematics and six factors related to Biology, Physics, Chemistry and English were sufficient to explain the total achievement variability. Thus, factors self-concept to the subjects, motivation to the subjects, interest to the subjects, trouble (anxiety) to the subjects, teaching-learning process and absenteeism explaining most of the achievement variations in five school subjects. Moreover, the result of general PCFA indicated that the factor named as family status (FS) that encompasses parentstudent communication, parent-teacher communication, availability of book at home, satisfaction in food available at home, mother educational level and father education level explained the higher variability for the overall achievement. This finding is in consistent with other studies ( [13]; [18]). The result of the multivariate multiple linear (single-level) regression analysis point to several interesting overall findings. The result indicated that the factors sex, school type, school facility (SF) which encompassed availability and satisfactoriness of school amenity, human resources, library, laboratory, equipment and academic counseling have significant impacts on achievements of the students in terms of all the selected five subjects. School volume (SV) that encompassed school size, class size, teacher workload and experience had a significant negative impact on academic achievements in terms of Biology and Chemistry. This may be due to the negative effect of school size, class size and teacher work load on academic achievement of students at school, as reported earlier ( [16]; [17]). The factor school facility (SF) that deals availability and satisfactoriness of the school instructional materials, school library, laboratories, amenities, academic counseling services and other school characteristics had significant positive impact in all five school subjects used as a measures of academic achievement. Family status (FS) which encompass parent-student communication, parentteacher communication, availability of book at home, satisfaction in food available at home, mother educational level and father education level had a significant positive impact on academic achievements in terms of Biology, Physics, Chemistry and English subjects as observed earlier ( [18]; [19]). This study was intended to identify some factors influencing the academic achievements of students' measured by five selected subjects (Mathematics, Biology, Physics, Chemistry and English) at secondary school level based on primary and secondary data. Accordingly, factor analysis, multivariate multiple linear regression and MVML multiple linear regression techniques on the five school subjects were employed. The factor analyses conducted in this study indicated that 4 or 6 factors (instead of twenty eight original observed variables or items) were sufficient to explain 77.4%, 76.7%, 78.8%, 68.6% and 73.4% the total variation in achievement for each separate PCFA of observed items related to Mathematics, Biology, Physics, Chemistry and English subjects, respectively. The factors self-concept, motivation, interest and trouble to the subject were the common factors explaining most of the variability of achievements in terms of each five subject, since these factors were appeared regularly in each separate PCFA. Moreover, six common factors were enough to explain about 64% of the variation using 34 originally observed variables in the generalized PCFA. The study revealed that the factors sex, school type, family status (FS) holding parents-student communication, parent-teacher communication, satisfaction in food available at home, availability of books at home, mother educational level and father education level, and school facility (SF) enclosing school instructional materials, amenities, library and laboratory facilities had statistically significant influence on achievements of students for the selected subjects. Moreover, school volume (SV) that covers school size, class size, teacher work load and teacher experience in teaching; interests to the subject, motivation to the subject, trouble to the subject and self-concept in school subjects have been significant factors ![the population measurements). S 1 2 = government school sample variance of students' academic achievement S 2 2 = non-government school sample variance of students' academic achievement = government school students sample mean of students' academic achievement and = non-government school students sample mean of students' academic achievement, is the critical value for 95% confidence level with standard normal distribution.](image-2.png "") ![the total sample for this study. Thus, using the above results, the following sample sizes for both school types (Governmental and Non-governmental) as proportional allocation by school type as a factor is: Sample for non-governmental schools).](image-3.png "(") ![v. The Contribution to the Total Sample VariancesIn applying the principal component to perform factor analysis, we have use, the sample covariance matrix S was used. Observe that =trace of the sample covariance matrix and = trace of sample correlation matrix, where,Researchers have no single agreement about selecting the required number of principal components. However, the best choices for researchers to fix the number of factors retained have been the proportion variance explained being at least 50-60% and the Scree plot test examining the graph of the eigenvalues by looking for the natural bend or break point in the data where the curve flattens out. The number of data points above the "break" is usually the number of factors to retain, although it can be unclear if there are data points clustered together near the bend([16;[21]]).vi. Rule of Thumb (Convention) ? Choose the number of positive eigenvalues of sample covariance matrix S and ? Choose the number of eigenvalues of sample correlation matrix R which are larger than 1. vii. Factor Rotation and Factor Scores Factor rotations are an orthogonal transformation of the factor loadings, as well as the implied orthogonal transformations of the factors. If is the matrix of estimated factor loadings obtained by any method, then](image-4.png "") ![the th i respondent/student for the th k factor retained, ij x = observation of the th i on the th j , j l ? = the principal component (factor) loading of variable j [15].](image-5.png "") ![error term ? has ( )? ? = ,Thus, the error terms associated with different responses may be correlated.Conceptually, we can let ( )](image-6.png "") ![explanatory (predictor) variables or factors= ZSetting the matrix of response (dependent) variables, ? and a matrix of fixed unknown parameter, ? and matrix of errors .](image-7.png "") ![trials (individual students) are uncorrelated ([9]; [21]). i. Method of Parameter Estimation In the model above ? and , =1, 2, 3,?, m, are unknown parameters. The ordinary least squares (OLS) estimates are found in a manner analogous to the uni-variate case. We begin by taking a single response solution as: ( )](image-8.png "") ![out from the regression model[9].](image-9.png "") 2![Graphs/Figures for Checking Model Adequacy PCFA of Biology Related Items PCFA of Physics Related Items PCFA of Chemistry Related Items PCFA of English Related Items](image-10.png "Appendix- 2 :") 120162![Figure 1 : The Scree Plots to Test for the Number of Factors Retained in the Generalized and Separate PCFA,](image-11.png "Figure 1 :- 2016 Figure 2 :") 31Year 201633Volume XVI Issue II Version IG )(School TypeSchool Name Comboni SOSSelected Subject (Hawassa, 2010) Students' Academic Achievement Maths Biology Physics Chemistry 35 35 35 35 Mean N 3.49 3.63 2.37 3.11 SD. 0.743 0.598 0.877 0.758 N 30 30 30 30 Mean 2.83 2.87 2.23 2.83 SD. 0.647 0.973 0.897 0.791English 35 3.68 0.471 30 3.07 0.827Overall Average 35 3.25 0.689 30 2.77 0.827Global Journal of Human Social Science -N303030303030AdventistMean2.902.772.632.772.802.77SD.0.8440.8170.9990.9710.8050.887N292929292929Mount OliveMean2.722.553.032.973.102.88SD..702.783.778.778.7720.763Non-N292929292929governmentBNBMean2.832.722.312.792.932.72SD..889.959.967.726.7530.859 3Students2 : Descriptive Statistics Student's Achievement in Ascending Order for the Overall Sample of 3Year 201635Volume XVI Issue II Version IG )(Global Journal of Human Social Science -4 : The Generalized Principal Component Factor Analysis (Hawassa, 2010)Accounted for 64.28%1Common Factors: Component 2 3 4 56CommunalityEigenvalues4.563.502.702.211.651.45Variations accounted for %18.2414.010.88.846.605.80Parent student communication.9020.841 33Separate Principal Component Factor AnalysisGeneralResponsesMathsBiologyPhysicsChemistryEnglishPCFA?Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization. 35ResponsesMathematicsBiologyPhysicsChemistryEnglishR20.720.740.680.680.71R2adj.0.640.670.610.640.65 36Parent teacher communication.8910.813Satisfaction in food at home.8530.767Availability of books at home.8400.750Year 201636G )(Global Journal of Human Social Science -s 1fostering problem solving skills, Bahir Dar University,29 (1):17 -29.Ethiopia. Educational Expert, Bureau of Education,19. Brian, H. and Russell, K. (2009). LANNA Tests andAmhara regional State, Ethiopia.the Prediction of Year 10 English and Mathematics21. Alan J. Izenman (2010). Modern MultivariateResults: Charles Sturt University, EducationalStatistical Techniques: Regression, Classification,Research, 19(1).and Manifold Learning.20. Alemayehu B., and Assaye A., (2010). An evaluationof grades 9 and 10 mathematics textbooks vis-à-visAppendicesAppendix-1Accounted for 77.38%Common Factors: Components1234Eigenvalues4.563.413.282.67Communality% Variations accounted for25.4218.9318.1914.84Need to do Maths well to get into the University.9050.832The teacher prepares well for Maths daily lessons.8940.813Need to do Maths well to get job.8730.782Learning Maths helps me in my daily life.8450.717Exam questions of Maths are standard.8390.726Teaching Maths covers the whole syllabus.8280.692Often study Maths in groups.9440. 907Maths is difficult to learn.9120.861No strength in learning Maths.8950.810Need lots of hard work studying Maths to perform well.8940.835Teaching method used by Maths teacher fits with the current curriculum.9330.880I am satisfied with the current curriculum of Maths.9210.857Maths need more time to understand.8860.809Maths is Boring.8330.729I usually do Maths well.8980.819Enjoy learning Maths.8740.784I have natural talent in Maths..7640.603Understand Maths quickly in class.6250.568Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization. 2International Journal of Educational Development,Year 201640Volume XVI Issue II Version IG )(Global Journal of Human Social Science -Students (Cronbach's ? =0.72, Hawassa, 2010) Accounted for 76.65% Common Factors: Components 1 2 3 4 5 Eigenvalues 4.13 3.52 2.34 2.17 1.63 %Variations accounted for 20.63 17.57 11.69 10.87 8.14 The teacher prepares well for Biology daily lessons .9146 1.55 7.75Communality 0.853Need to do Biology well to get into the Preparatory or University.9070.847Need to do Biology well to get job.9020.820Learning Biology helps me in my daily life.8970.833Teacher is efficient and skilled while teaching Biology.8640.754I usually do Biology well.9490.922Understand Biology quickly in class.9110.864Enjoy learning Biology.9000.844s 3Students (Cronbach's ? =0.71, Hawassa, 2010)Common Factors: ComponentsAccounted for 78.80%123456Eigenvalues3.903.753.642.861.601.58Communality% Variations accounted for17.7517.0616.5513.017.277.18Need to do Physics well to get job.8980.835Need to do Physics well to get into the Preparatory or University.8710.789Teaching Physics covers the whole syllabus.8500.784Learning Physics helps me in my daily life.8370.738I have natural talent in Physics..7940.676I usually do Physics well.9670.951Understand Physics quickly in class.9650.952Enjoy learning Physics.9640.948Physics need more time to understand.9560.923Teacher is efficient and skilled while teaching Physics.9410.913I am satisfied with the current curriculum of Physics.9400.933Often study Physics in groups.9290.502Need lots of hard work studying Physics to perform well.9110.888Physics is Boring.9770.961Physics is difficult to learn.9730.955No strength in learning Physics.9470.899Physics teacher is often late for class-.7300.609Student get at least a onetime Physics homework /assignments/ class works per week.6970.565Physics teacher often absent from class-.6880.596Exam questions of Physics are standard.7780.641Teaching method used by Physics teacher fits with the current curriculum.6480.547The teacher prepares well for Physics daily lessons.5560.573 5Accounted for 73.43% Eigenvalues1 3.66Common Factors: Components 2 3 4 5 3.59 3.50 2.70 1.456 1.25Communality% Variations accounted for16.6516.3115.9312.286.595.67Need to do English well to get into the Preparatory or University.8620.758Learning English helps me in my daily life.8570.751The teacher prepares well for English daily lessons.8330.738Need to do English well to get job.8270.741Teacher is efficient and skilled while teaching English.7840.683Exam questions of English are standard.8500.774Need lots of hard work studying English to perform well.8450.732I am satisfied with the current curriculum of English.8440.783Teaching method used by English teacher fits with the current curriculum.8430.797English need more time to understand.7430.577 © 2016 Global Journals Inc. (US) ## Acknowledgements First, and foremost, I would like to thank my God for giving me the opportunity to allow me learning Statistical and doing researches in every fields of sciences, and for being with me in all aspects elsewhere. I would like to gratefully and sincerely thank Dr. Ayele Taye, for his invaluable comments and suggestions that contributed to the successful realization of this study. He devoted his time to provide me insightful feedback on drafts of this manuscript and challenged me to become a better researcher and scholar. I would like to acknowledge the efforts of the School of Natural and Computional sciences and all my colleagues for their scholarly input I got through the discussions. My special thanks also go to the staff members of the Department of Education and Capacity Building at Hawassa City Administration, school principals and home room teachers of all secondary schools in the city for their positive and genuine cooperation provided to me during data collection and those selected grade 10 students who gave me the information and data used in this study. I owe tremendous amount of gratitude to my parents, friends and relatives; specifically, I feel greatly indebted to my brother Awol Assen, my friends Bereket Adem, Hussien Ali and Mebrat Seid who were never failed to give me support, continuous encouragement and appreciation for my studies. 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