CROSS VALIDATION OF A MULTIVARIATE PATH ANALYSIS OF
PREDICTORS OF HOME SCHOOL STUDENT ACADEMIC ACHIEVEMENT

Terry J. Russell
School of Social Work
University of Pittsburgh
119 Gene Drive
Pittsburgh, Pennsylvania 15237

Keywords: Homeschooling, home schooling, home education, academic achievement.

Are home school students getting a good education? The literature reviewed for this study provides evidence that home schooled children, as a group, are getting a good education and are performing academically on a par with their public school peers. Building on a growing body of research, this study seeks to answer the next logical question, “Within the home schooling population, what factors contribute to academic achievement?”
This is an exploratory study, and as such does not attempt to test a hypothesis. The purpose is to measure the effects of several previously measured variables upon academic achievement. Using the data from a previous study (Wartes, 1990), nine questions can be explored.
Of the nine questions to be posed and formed into variables, five involve pre‑existing demographic factors, as follows.
1. Does family income have an effect on student academic achievement?
2. Does the parent’s level of education have an effect on student academic achievement?
3. Does the home school student’s grade level have an effect on student academic achievement?
4. Does the number of years a student has been home schooled have an effect on student academic achievement?
5. Does whether the student was previously public schooled, private schooled, or always home schooled have an effect on student academic achievement?
The remaining four variables are derived from questions about how the home schooling is administered, including training which the parents may have received prior to or during the home school experience.
6. Does whether the parent has had training in home schooling have an effect on student academic achievement?
7.  Does the amount of religious content incorporated in the curriculum have an effect on student academic achievement?
8.  Does the amount of structure used by the parent in home schooling have an effect on student academic achievement?
9.  Does the number of hours per week spent in home schooling have an effect on student academic achievement?
As stated above these are not hypotheses to be tested, but questions to be addressed in an exploratory analysis. The objective of this study is to answer these questions based on the data from the Washington Homeschool Research Project (Wartes, 1990).

A Review of Home Schooling Research

There is evidence that home schooled children academically are doing as well as or better than their more traditionally educated peers. A body of research, documented in detail by Ray (1988), indicates home school students do as well as or better academically than their public school counterparts. Some questions remain as to whether this is because home schooling provides a better education or because of some rival explanation, such as whether the students of home schools tend to have certain other advantages. As Ray concludes, “It is possible to hypothesize that the home school treatment is a causal factor of higher achievement,” but the research to date cannot be said to prove such hypotheses” (p. 26, italics added). The fact remains, as reported in U.S. News and World Report, that home school students have been shown academically to compare favorably with public school students (Toch, 1991).

Review of the Wartes Study

For the purpose of this paper it was decided to perform a follow up of the Washington Homeschool Research Project (Wartes, 1990). This data set consists of 877 subjects, home school parents, who completed a survey during 1987, 1988, and 1989. A student is considered a home school student if 75% or more of what the family considers to be schooling is provided by or conducted under the supervision of the parent(s). The original study was limited to bivariate analysis in which correlations were computed and reported. As such, no attempt was made to test for any cause‑effect relationships, or to control for any third variables that might have an intervening or moderating effect on a bivariate relationship. Wartes presented results on 17 topics. In general, any statistically significant correlations were low.
The 17 topics presented by Wartes were broken down into three categories. Descriptive topics included single‑parent families, those planning to discontinue home schooling, previous type of schooling, parent’s education level, handicapped students, age distribution, and whether the parents had took a course in home schooling which was required in some cases and voluntary in others. Findings included that the two percent who were home schooled by a single parent scored average to above average on the Stanford Achievement Test (SAT); there was no relationship reported between test scores and intent to continue or discontinue home schooling; no significant differences were reported between those who were previously public, private, or home schooled only; a significant but weak relationship was reported between parents’ income and parent’s education level; three percent of the students were regarded by their parents as moderately to seriously handicapped; the students ranged in age from 5 to 18, with 9.9 being the average; and about 35% of the parents had taken a class in home-based education, almost half of which had not been required to do so.
Four of the 17 topics had possible implications for public policy. A weak relationship was reported between parent’s education level and the student’s SAT scores, but as a group the children of parents with only a twelfth grade education scored somewhat above the national norm. No significant relationship was reported between SAT scores and contact with a certified teacher. Little or no relationship was reported between either the level of structure or the hours per week spent home schooling and the student’s SAT scores. Little or no relationship was reported between student’s grade level and SAT scores.
Other topics addressed in Wartes’s study included length of time home schooled, family income, gender comparisons, success in previous conventional schooling, and relationships involving religion. No relationship was reported between length of time home schooled and student’s SAT complete battery score. A weak relationship was reported between family income and student’s SAT scores. No gender differences were reported for the SAT complete battery score, although females did better in the Total Language subscale and males did better in the Science subscale. A positive relationship was reported between student’s SAT scores and their previous success in conventional schooling. No relationship was reported between student’s SAT score and the degree of religious content used in home schooling.
One of the limitations cited by Wartes is the bivariate analysis used, which is directly addressed by the present study based on multivariate path analysis. Wartes noted that student academic achievement is the result of a complex set of factors, but regretted the inability to consider several variables at a time. By considering numerous separate bivariate relationships the chance of type I error is increased.
Wartes addressed this issue by relying on a .01 probability of error rather than .05 as an indicator of statistical significance. This means of compensating for type I error is not without merit, but a better solution is to use multivariate analysis.
Although type I error is managed much better by multivariate analysis, it is still a threat. With a large data set, such as this, cross‑validation goes even further toward confidence that significant findings are valid and not the result of type I error. Additionally, path analysis, or causal modeling, provides a visual representation of the cause and effect relationships between several variables. In a fully recursive causal diagram all the causal influences are assumed to be in one direction, illustrated by diagramming the variables from left to right in the order of the assumed causal direction (Bohrnstedt & Knoke, 1982).

Method

The availability of such a large data set (N=877) provided the ability to evaluate reliability using split samples for cross‑validation (Kleinbaum, Kupper, & Muller, 1978). It was decided that the full data set would be randomly split and an exploratory model tested on the first random sample of the full data set, subset A. Then, based on the results, a reduced model would be tested on the other sample, subset B. Several variables are included in this study, increasing the chance of type I error, but the split samples procedure would provide a measure of reliability so that “taking advantage of chance” was not an undue problem (Nunnally, 1967).
The plan then was to identify non‑contributing variables with subset A for elimination in the analysis of subset B. The reduced model would be tested and the findings of the second analysis used to cross‑validate the findings of the first. Once this model was trimmed and cross‑validated, it then would be applied to the full data set in order to measure the effects of the remaining causal variables upon the dependent variable using all of the available data. Since the full data set was split randomly into sample A and sample B, any differences between the two groups are random and do not reflect bias. A comparison of means and standard deviations for the full data set and for the two subsets revealed no substantial differences.

Respondents

The subjects are those home schooling parents who conformed to the state requirement for testing and who voluntarily completed the questionnaire. Subjects were not randomly selected, the entire population of Washington state home schoolers who took the SAT test battery were asked to participate. Those who participated were self‑selected, allowing selection bias to threaten the generalizability of the findings.
In the original study, Wartes (1990) discussed this at length. He compared the respondents with non‑respondents and reported that there were some cases in which differences between these groups were statistically significant. “Where a difference existed, it always favored the group that had returned more questionnaires but the magnitude was usually small” (p. 86). He stated that the conservative action was to assume that the respondents were not typical, and he concluded that this sample may not be representative of the population of Washington home schoolers who conformed to the legal requirement by being tested. Wartes’s findings and conclusion do not indicate that the sample is not representative, they indicate that the representativeness of the sample is at least somewhat questionable. Therefore the external validity, or generalizability of this study to home schoolers not in the study, is threatened somewhat.
The average age of the home school students in the study was 10 years with a standard deviation of 2.5 years; on average they were in fourth grade. They were evenly distributed between male (48%) and female (52%). The average annual family income was about $30,000 with a standard deviation of $11,800; 94% of which are one income families. The parent primarily responsible for home schooling had an average education level of 14 years, with a standard deviation of 2 years. The vast majority plan to continue home schooling the next year, with 77% reporting they definitely will, and only 7% that they definitely will not home school the next year.
The previous school performance of children was predominantly average to above average, as reported by the parents. This would have been included as a variable in the analysis but 74% of the respondents did not answer the question, making it unacceptable as a variable. Much of this is due to the fact that 30% of the students in the study were always home schooled and have no previous class ranking. This still leaves an inordinate amount of missing responses, which can be speculatively attributed to parents not knowing what there child’s previous class ranking was. Of the 26% who did respond to the question, the largest group (41%) were in the top 20% of their class, 20% were in the next to the top 20%, and 27% were in the middle 20% of their class.
The respondents were predominantly Christians in non‑mainline Protestant denominations, 40% referred to themselves as conservative evangelicals and another 42% as non‑denominational Christians. This matches other studies which show that home schoolers tend to be religious conservatives who avoid membership in mainline denominations (Mayberry, 1988).
The average number of years the student had been home schooled was 2.7 with a standard deviation of 1.5 years. The type of previous education was fairly evenly distributed between three groups, previously public schooled (39%), previously private schooled (28.5%), and always home schooled (30.8%), with 1.5% reporting “other.”

Measures

The Dependent Variable: Academic Achievement
The dependent variable, academic achievement (complete battery score), was measured in Normal Curve Equivalents (NCEs). These NCEs were derived by converting the individual national percentile score taken from the Stanford Achievement Test series and based on 1986 norms. Although percentile scores are widely used and are familiar to the general public, converting them to NCEs is necessary to provide an equal interval measurement scale suitable for the statistical analysis used here (Wartes, 1990).
This series consists of the Stanford Early School Achievement Test for grade K (The Psychological Corporation, 1986), the Stanford Achievement Test for grades 1‑8 (The Psychological Corporation, 1986), and the Test of Academic Skills for grades 9‑12 (The Psychological Corporation, 1986). “The same score scale ties SAT with the Stanford Early School Achievement Test (SESAT) and the Stanford Test of Academic Skills (TASK),” providing a measurement scale that continues across grade levels from kindergarten through high school and into community college (Anastasi, 1976, p. 408). Table 1 provides a chart showing the comparable values of percentile scores and NCEs as used in this study, as well as the amount of NCE change for each percentile score change. Note that the measured change in NCE for a one percentile change varies from .5 at the center of the distribution to 5.7 at the extremes at either end of the distribution. This reflects the difference between percentile scores and an equal interval scale, and underscores the necessity of converting the scores for statistical analysis.
Regarding content validity, Anastasi (1976) states that the development of the SAT demonstrates outstanding technical quality, and that “unusual
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Table 1. Conversion chart for normal curve equivalents (NCE) and percentile rank.

ingenuity is evidenced in the design of objective items to measure a wide variety of complex cognitive functions” (p. 408). Reliability coefficients are computed for each of the subject tests for each of the grade levels. The majority of these numerous reliability coefficients are in the .80s and .90s, with a few in the .70s. “On the whole, the reliabilities, except as noted, are entirely satisfactory, especially in view of the fact that each is calculated for a sub‑test within a single grade, where both the shortness of the tests and their restricted range have a depressing effect on reliability coefficients. The reliability of any total battery would probably be in the area of .95” (Noll & Scannell, 1972, p. 277).

Predictor Variables
Most of the variables were measured by objective responses. Family income was measured in $5,000 increments from “0 to $4,999” to “$50,000 and above.” Parent’s education was the actual number of years of education completed by the parent primarily responsible for the home school instruction. School grade was the grade level of the student. Years home schooled was the actual number of years the student had been home schooled. Hours per week was the average number of hours spent each week in home school instruction.
Two variables were subjective responses on a seven point scale. Religious content was the parent’s self‑rating with “1” being no religious content and “7” being a very high degree of religious content. Structure was the parent’s subjective rating also, with “1” being very little structure and “7” being very highly structured.
Parent training was a dichotomous variable, either the parent had taken a class, coded “0,” in home-based education or not, coded “1.” In Washington state, home school parents with less than one year of college education are required to take a class in home-based education. Of the total sample 35% took such a class, either because they were required or because they chose to. This means that two variables, parent training and parent’s level of education, are confounded in a way that one’s effect could mask the effect of the other. Entering all variables in the regression analysis tests each variable while controlling for the others, so the result indicates each variable’s effect while eliminating the masking effect of the other.
The remaining variable, type of previous education, was categorical but was transformed into three dummy variables; previously home schooled only, previously public schooled, and previously private schooled. This single categorical variable was treated as three dichotomous variables in order to allow their use in the analysis. This resulted in a total of twelve variables, arranged chronologically from left to right in a fully recursive path diagram.

Results

The path analysis of subset A was fully recursive, all causal paths were assumed to be in one direction and all paths in the model were tested since no particular hypotheses were being tested. Using the Statistical Package for the Social Sciences (SPSS, Inc., 1990), the dependent variable was regressed on all independent variables in order to test the direct effect of each variable on the outcome. A path coefficient of at least .05 was considered substantial enough to qualify the variable for inclusion in analysis B for cross‑validation. Variables with a path coefficient of at least .05 were regressed in turn on all prior variables until all potential paths, direct and indirect, were detected.
The second step of the analysis plan was to test the model derived from the first step. The path analysis of subset B assumed the paths identified in the path analysis of subset A. Only these paths were tested. As a final step the cross‑validated model, defined in step 1 and tested in step 2, was applied to the full data set. For improved prediction purposes the path coefficients then were computed and reported using the full data set.

Analysis of Subset A
The results of the analysis of subset A are intermediate results and not to be taken out of the context of the cross‑validation process. Variables at this stage of the process are either eliminated as causal factors or remain in the model as possible direct or indirect causal factors. The following results of analysis A then are not offered as findings but as intermediate results.
The largest potential direct effect on academic achievement found in the analysis of subset A was parent’s education with a path coefficient of .23, (p < .01; see Figure 1), followed by degree of structure (path coefficient of .22, p < .01), and hours per week (path coefficient of ‑.22, p < .01). Hours per week, interestingly, had a negative direct effect. This seems to indicate that an increase in the hours per week causes a slight decrease in test scores, or that slower learners need more time, but at this point it was premature to speculate since the intermediate results remained to be tested on data subset B.

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Figure 1. Results of path model analysis for subset A showing all path coefficients greater than .05, and path coefficients greater than .10 are represented by darker lines.

      Of the remaining direct effects, the type of previous education being private school had a negative path coefficient next highest to the three just mentioned, ‑.20. As a dummy variable, this was coded as a one (1) for those who were previously
private schooled and all others were coded as zero (0). Thus, students who were previously private schooled scored slightly lower than others as a group, in the pre‑cross‑validated analysis A.

The two remaining direct variables are having always been home schooled, with a small path coefficient of .08, and student’s current grade level, with a path coefficient of .05. These are low and are not statistically significant, but it was decided at the outset that any path coefficients over .05 would remain in the model.
Only religious content was isolated as having no direct effect on any other variable. This verified findings in the original bivariate study that indicated no relationship between religious content and test scores, as such it would be eliminated from the model in the analysis of data subset B.

Analysis of Subset B
The analysis of data subset B partially replicated the findings from the analysis of data subset A (see Figure 2). Analysis A resulted in six direct effects on

complete battery score with path coefficients of at least .05. In analysis B the dependent variable was regressed just on these six variables. Only two paths were replicated, parents education with a path coefficient of .18, p < .01, and structure with a path coefficient of .13, p < .05. These direct effects are cross‑validated by analysis B, but the best measurement of the size of their effect will come from the full data set.

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Figure 2. Path diagram for the analysis of data subset B showing all effects with path coefficients greater than .05, and path coefficients greater than .10 are represented by darker lines.

In analysis A there were several indirect effects, in analysis B there are only two indirect effects.
Family income had a path coefficient of .05 on structure, and always having been home schooled had a path coefficient of ‑.08, neither of which is statistically significant. The product of either of these path coefficients and the .13 path coefficient of

structure results in a minute indirect effect upon complete battery score. These indirect effects are validated but, again, the best measurement of these indirect effects will be derived from the analysis of the full data set.
Direct effects not replicated in the analysis of subset B include school grade level, previously private schooled, previously public schooled, and hours per week. All but hours per week were small and not statistically significant, so it is
not surprising that they were not replicated. Hours per week, however, had a path coefficient of ‑.22, p < .01 in analysis A, in analysis B it was a statistically non‑significant .02. The finding in analysis A was apparently due to random error, created by chance when the full data set was randomly split.
Summary of results

Having defined a path model in the analysis of data subset A, tested the model in the analysis of data subset B, the cross‑validated model was applied to the full data set in order to get the best estimates of the causal effects of the remaining independent variables on the dependent variable. This consisted of two variables with direct effects on complete battery score, parent’s education level and structure, and of two variables with indirect effects through the structure variable, family income and always having been home schooled (see Figure 3).

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Figure 3. Final path diagram: the cross-validated model applied to the full data set in order to compute the best measurement of each path coefficient.

      The best predictor of home school student achievement in this study is parent’s education level. Although this is a statistically significant finding, it is helpful to measure the effect in real terms. A one standard deviation change in the independent variable results in a one standard deviation change in the dependent variable multiplied by the path coefficient. Dividing both of these figures by the standard

deviation of the independent variable yields the predicted change in the dependent variable for a one unit change in the independent variable. Therefore, based on this full data set, a one year increase in parent’s education level results in an increase of 1.9 Normal Curve Equivalents of the percentile increase on the SAT complete battery score.
From Table 1 it can be shown that a 1.9 NCE change can correspond to any percentile change from a high of 3.8 percentiles near the center of the distribution to a low of .3 percentiles at the extreme ends of the distribution. Additionally, at the 25th and 75th percentiles a 1.9 NCE change reflects a 2.7 percentile change. The students tested in this study had a mean average NCE of 59.3, which corresponds to the 67th percentile. At this average NCE level of 59.3 a 1.9 NCE change corresponds to a 3.2 percentile change. Therefore it can be said that for the students used in this study with an average SAT score, a one year increase in parent’s education level resulted in a 3.2 percentile increase on the SAT.
The measurement of the effect of structure on complete battery score is less meaningful to compute because the independent variable, structure, was measured by a subjective self‑rating on a scale of one to seven of how much structure the parent reports using in home schooling. The inherent problem with subjective self‑ratings is that what one person considers to be high another might consider to be low. Additionally,  subjective responses do not provide a true interval scale measurement of the actual construct attempted to be measured. This acknowledged, it stands that one point on an equal‑interval seven point scale is 14%. If this is applied to the subjectively measured variable structure, with a standard deviation of 1.23, a path coefficient of .09 between structure and complete battery score corresponds to an increase of 1.4 Normal Curve Equivalents to the SAT complete battery percentile score for each one unit increase, on a 7 point scale, in the amount of structure reported by parents.
Referring to Table 1, it can be seen that a 1.4 NCE change corresponds, depending on where it falls in the distribution, to any percentile between a high of 2.8 percentiles and a low of .25 percentiles. The percentile change corresponding to 1.4 NCE at the 25th and 75th percentile levels is 2. At the mean average for this data set, the 67 percentile, a 1.4 NCE change corresponds to 2.3 percentile points.
The direct effect of always home schooled on structure (‑.13,  p < .01) provides the path for a small indirect effect on complete battery score. Always home schooled had a ‑.35 unit effect on structure. If each measured unit of structure is considered to be 14% of the total, a ‑.35 unit change represents a change of ‑4% in amount of structure used as reported by parents. The amount of this indirect effect can be computed by multiplying the product of the two direct effects by the standard deviation of complete battery score (19.3) divided by the standard deviation of always home schooled (.462). This results in an indirect effect of always home schooled on complete battery score of ‑.42, or just less than a one‑half of one Normal Curve Equivalent to a percentile drop in SAT complete battery score for those who have always been home schooled versus any other type of previous education.
Again referring to Table 1, it can be seen that a ‑.42 NCE change can correspond to any percentile change from a high of ‑.84 at the 50th percentile level, ‑.6 at the 25th and 75th percentile levels, ‑.7 at the mean average 67th percentile level, and a low of ‑.07 at the extreme percentile levels of 1 and 99.
The only other indirect effect cross‑validated in this analysis was that of family income through structure. This effect was small and not statistically significant in the cross‑validation analysis. When the final model was applied to the full data set the path coefficient was only ‑.01 and should be discounted as having any measurable effect on structure or complete battery score in this data set.

Discussion

Most of the variables in this study were not found to be predictors or causes of home school student academic achievement. The nine questions posed at the beginning of this study can be answered, at least for the studied sample, from the data as follows.

Does family income have an effect on student academic achievement?
Family income does not have a direct effect on the complete battery score of the home school students in this study. In the final application of the cross‑validated model using the full data set, family income had no measurable effect on structure and therefore no indirect effect on complete battery score. These data indicate that family income is not a predictor of academic achievement as measured by the Stanford Achievement Test.

Does the parent’s level of education have an effect on student academic achievement?
Of the nine variables considered in this analysis the parent’s level of education is the single best predictor of academic achievement for home school students, although the effect is not large. Based on this full data set a one year increase in parent’s level of education predicts nearly a two (1.9) NCE increase in home school students’ complete battery score on the SAT, which corresponds to a 3.2 percentile increase for the average student in the study sample.
It should be remembered that these data do not allow for comparisons to public or private schooled students. It may be that public and private schooled students experience a similar relationship between parent’s education level and student’s academic achievement. Although this analysis found parent’s education level to be a predictor of some academic achievement, there is no evidence from this study that this is unique to home school parents.

Does the student’s grade level have an effect on student academic achievement?
The small direct effect found in data subset A was not cross‑validated in the analysis of data subset B, neither were any indirect effects of student’s grade level on academic achievement. Based on this data there is no evidence that students of lower or higher grades respond better or worse to home schooling in terms of academic achievement as measured by the SAT.

Does the number of years a student has been home schooled have an effect on student academic achievement?
In the analysis of data subset A the number of years a student has been home schooled showed some promise as an indirect effect through two other variables. Neither of these other variables was cross‑validated as effects, however, so this study finds no direct or indirect effect of years home schooled on academic achievement for home school students as measured by the SAT.

Does the previous type of education have an effect on student academic achievement?
Always having been home schooled did not have a direct effect on academic achievement in this data set. However, always having been home schooled did have a significant cross‑validated indirect effect on academic achievement through the amount of structure used. The amount of structure reported to be used has a positive effect on academic achievement, but parents of students who have always been home schooled tend to report using less structure in home schooling their children than do other parents of home schoolers.
The negative indirect effect of having always been home schooled predicts less than a one‑half of one NCE decrease on the complete battery score of the SAT. Although statistically significant this effect might be considered too small to concern home schoolers and those who legislate policy regarding home schooling, especially since structure was measured by a subjective self‑rating. Parents of students who have always been home schooled may be advised, on the basis of this finding, to guard against being too unstructured in their home schooling. This should not be considered an issue to be regulated, however, since the indirect effect is small and these students as a group are performing above average compared to public and private school students.

Does whether the parent has had training in home schooling have an effect on student academic achievement?
The only effect of the training class found in the analysis of data subset A was an indirect effect on complete battery test score through a positive direct effect on structure. This was not cross‑validated in the analysis of data subset B, however, thus eliminating the indirect effect on the complete battery test score. It should be remembered that this applies to the effect of this particular training class on this particular set of home schoolers, and does not apply universally to any or all other opportunities for additional training.

Does the amount of religious content incorporated in the curriculum have an effect on student academic achievement?
With a mean of 4.8 and a standard deviation of 1.5 there was a good distribution of responses with the average tending to be somewhat more religious than secular. This variable did not have a direct or indirect effect on academic achievement of the home school students in this study.

Does the amount of structure used by the parent in home schooling have an effect on student academic achievement?
Structure is one of the two variables in this study that have a significant direct effect on home school student academic achievement. This effect was undetected in the original bivariate study, but when controlling for other variables in the study it is shown to be significant. Because the amount of structure used was measured by the respondent’s subjective self‑rating, the effect of structure cannot be interpreted as a firm measurement. It is best generally to conclude that structure had at least some small, positive effect on academic achievement as measured by the SAT scores of the students in this study.
The other role of structure in this model is that of path intermediary for the indirect effect of always home schooled on complete battery score. Tending to use less structure, or to report using less structure, is a small direct effect of always being home schooled, as discussed above, which in turn has a small negative effect on SAT score. This indicates that parents of students who have always been home schooled might eliminate this possible negative effect by using more structure.

Does the number of hours per week spent in home schooling have an effect on student academic achievement?
The number of hours per week spent in home schooling varied greatly. The average was 15 hours per week and the standard deviation was nearly half that at 7.3. The responses ranged from zero (0) to 45 hours per week and were well distributed between these two extremes.
Based on the finding of this analysis the number of hours spent per week in home schooling does not significantly affect the student’s academic achievement. Any effect of this variable may have been diluted by extraneous variables not measured in this study, which would enable some students to learn at a faster or slower pace than others. If students with certain advantages require less time to learn the same amount as other students, then the effect of hours per week cannot be properly tested without controlling for those advantages.
From this study it cannot be concluded whether or not the hours per week spent in home schooling have any effect on academic achievement for students with the same ability and opportunity. An attempt to partially accomplish this could have been made in this study by including as a variable how well the students did in their previous experience. Unfortunately this variable, as discussed earlier, could not be included because of the excessive number of missing values. Future studies would do well to include this as an appropriate control variable.
With this in mind, it is interesting to note that it is not the amount of time spent home schooling but the degree to which that time is structured, that was found to have a positive effect on SAT scores for the students in this study sample. Home schoolers might be advised on the basis of this data to be more concerned about structure than about the number of hours.

Limitations

A major limitation is the inability to make direct comparisons to public and private schooled students on the variables included in this study. This was not the intent of the study, since it was not possible with the available data, but such a comparison would provide a valuable addition to the present study. The outcome variable, Stanford Achievement Test scores, is based on a national norm. This provides a comparison across different types of education regarding how the home schooled students in this study are doing academically based on national norms. What this study cannot do is determine if the effects of the variables in the model differ from the effects of these same variables on academic achievement of non‑home schooled students. In other words, the variables with significant effects on test score could very well have the same or even greater effect for public or private schooled students. Future studies of home schooling should test models using data and variables that apply to public and private school students as well as to home school students in order to determine if there are differences in how students in the different types of schooling are affected.
The use of the SAT as the sole measurement of the outcome variable, student academic achievement, presents another limitation. The SAT measures some aspects of student academic progress, but alone it does not have sufficient content validity to serve as the sole operational definition of educational progress and success. There may be positive and negative effects of home schooling which are not measured or intended to be measured by the SAT.
A limitation already discussed is the self‑selection of the respondents in this study. The external validity of this study is threatened because there is some doubt about the representativeness of the sample. The findings of this study apply to the sample from which the data came. The ability to generalize these findings to all home schoolers is dependent upon how well the respondents represent the population of home schoolers in Washington state and the nation.

Conclusion

In general, this cross‑validated multivariate path analysis supports the findings of the original bivariate analysis done by Wartes of the Washington Homeschool Research Project. By not finding large predictors within the data, the results of both types of analyses provide greater evidence that none of the home schooling characteristics considered assert powerful influence on the success of home schooling.
While it seems prudent to require home school students to be examined by some standard of academic success, and to continue researching this alternative form of education, this and other research indicate that there is no reason to doubt that home school students are receiving a good education.

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