Self-Determination Theory Involving Principal Component Analysis. Work Presented to Ivan Ivanov
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1 Self-Determination Theory Involving Principal Component Analysis Work Presented to Ivan Ivanov Carolane Radman and Paul Hankewicz 5/16/2012
2 2 TABLE OF CONTENTS Introduction.p.3 Theory.p.3-4 Statistics.p.5 Principal Component Analysis.p.6-7 PC Analysis of a survey...p.7-8 Data Analysis of the Covariance Matrix.. p.9-11 Comparison..p.12 Conclusion.p.13 Bibliography.p.14
3 3 INTRODUCTION This report is a statistical analysis of a survey conducted on psychology students. With the use of the principal component analysis, one can determine the degree and type of motivation of the students using the self-determination theory. The collected data will show whether or not the experiment shows a relationship with the motivation and information contained in the data. This can be completed by a method for computing the amount of accountable data expressed by the primary principal component of the covariance matrix. This true eigenvector-based multivariate analysis is known as principal component analysis. THEORY The self-determination theory (SDT) is a theory based on motivation and personality of humans. It relates to the interest, the concern, and the tendency behind a person s choice for a determined activity. The researches on this theory started in the 1970s. A more developed theory involved two aspects of motivation: intrinsic and extrinsic. However, these new aspects were accepted only in the 1980s as part of the theory. Intrinsic motivation implies that the motives are due to the interest, the fascination, the satisfaction of doing something or the personal development. On the opposite, extrinsic motivation implies that something is done based on the idea of an award at the end of the activity (external source) like fame or money. The SDT calculates the degree of determination of an individual based on their choices. After a couple of years, three psychological needs to motivate the self were added by Deci and Ryan in order to improve the definition of the theory and the comprehension and calculation of it. The three needs are competence (control on a situation and the outcome), autonomy (act for its own self), and relatedness (desire of interaction and connection for the approbation of others). The needs have an impact on wellness and are defined in relation to the social environment. It refers on how factors like social and cultural aspects change the degree of motivation of people. SDT is said to be an organismic dialectical approach and it includes five mini theories: the Cognitive Evaluation Theory (CET), the Organismic Integration Theory (OIT), the Causality Orientation Theory (COT), the Basic Psychological Needs Theory (BPNT), and the Goal Contents Theory (GCT). The first one, CET, is related to intrinsic motivation, the satisfaction of doing something only because the person finds it interesting. CET is related to two psychological needs: competence and autonomy because it involves talent/passion towards something and the desire to do it without having anyone to ask for it. Therefore, it is critical in education, sports, arts, etcetera since those domains rely on the talent (competence) and the autonomy to do it; no one will tell you what to do on the field, how and what to draw or when and how you should study. The second one, OIT, is related to extrinsic motivation. It is related to a behavior that is instrumental, meaning that it needs a concrete reward to function/to have something done. There are four forms of instrumentality which are external regulation, introjection, identification, and integration. External regulation is based and performed only on a reward (no autonomy is involved since it needs a possibility of gaining something to do it). Introjection is involved with the ego; an activity is done to
4 4 show people its abilities and talents (approbation and even admiration of others). Identification implies when there is a goal to accomplish so that it becomes personally important. Finally, integration shares some aspect of intrinsic motivation, but it is still considered extrinsic because it is related to the achievement for the own believes and personal needs. To summarize, the OIT explains that the more internalized (focus on the self) the motivation is, the more autonomous the person will be. Therefore, it supports more the autonomy and the relatedness than the actual competence since it involves the perception that others will have and the inner satisfaction of doing something (goal to achieve). The third one, COT, describes the differences in people for the orientation or the regulation of the environment and the behavior. It involves three types of causality orientations: autonomy orientation (act out of interest in and valuating of what is occurring), control orientation (focus on rewards, gains, and approvals), and impersonal orientation (anxiety concerning competence). The fourth one, BPNT, elaborates the concept of psychological needs and their links to health and well-being. It argues that the three psychological needs predict the mental well-being of a person. For this mini theory, the three needs are essential and, if they are affected, it will have an impact on the mental of the person. BPNT is all about the psychological aspect of the human. The fifth and last one, GCT, demonstrates the distinction between intrinsic and extrinsic on the wellness. Goals are seen as a basic need for satisfaction, a satisfaction that is different than the wellbeing. The goals are extrinsic because they involve financial, appearance and popularity aspects which are contrasting with the intrinsic that involve community, close relationships and personal growth. Extrinsic goal are greater in wellness whereas intrinsic goals are greater in well-being. In his research on SDT, Deci looked at the effect that a reward would have on someone that has an intrinsic motivation. Because other experiments were done on humans and animals, Deci was able to focus on only two aspect of the addition of a reward. In total, he did three experiments to justify or anneal his two hypotheses. Deci s first hypothesis was that if a reward was added, the intrinsic motivation would decrease and the second one was if a different type of reward would, in opposite, increase the degree of intrinsic motivation. The first and second experiments were testing the first hypothesis in the difference that experiment one was done in a controlled environment (laboratory) and that experiment two was done in a natural situation. The experiment was done on a group of psychology students that were separated in two and that were assigned sessions. On some of the sessions, only intrinsic motivation was involved, but in others, a reward was added to the activity. It was found that monetary reward decreased to degree of intrinsic motivation. In the third experiment, money as reward was replaced by verbal encouraging talk. This experiment related to the second hypothesis in trying to show that if the reward was the social approbation, the degree of intrinsic motivation would increase. The experiment was done the same way as the first experiment and it confirmed that the degree of intrinsic motivation increases when the external reward is verbal praise. The self-determination theory relies on psychology and statistics to try to understand, to evaluate, and to predict the reactions and actions of people. It is performed with the help of a survey that will calculate a certain degree or type of motivation. The results are then analyzed by a statistics method which is the principal component analysis. This gives a perspective and a method to determine whether or not the person was motivated and how. It is centered on the persistent positive features that humans can demonstrate by the efforts input and the commitment of an activity.
5 5 STATISTICS The study of analyzing collected and organized data is known as statistics. It encompasses various aspects of experimentation through the design of surveys, tests and other qualitative and quantative experiments. A person who has adapted in this field and can create an analysis with ease is known as statistician. It can be developed through the practice of understanding raw data in various fields such as biology, business, chemistry, and physics or honed in the discipline of mathematical statistics. However, due to its broad use in scientific applications, it is considered to be a distinct mathematical science as opposed to a branch of mathematics. In addition to this, most of statistics is non-mathematical, because the data collection is organized in such a way that many conclusions can be drawn depending on the reasoning and justification of the analyst. However, the quality of the data can be improved by survey sampling which is a process of selecting a grouping (or sample) of elements from a certain population in order to conduct a survey. Another way of increasing the quality is through the design of experiments, which are various ways to gather information in a certain type of experiment. In statistics, it is the control variable that can alter the quality. A control variable is an element that is kept constant within an experiment to ensure precision and accuracy within the results. In contrast, an independent variable is a value that can be manipulated or changed resulting in various dependent variables to change by cause and effect. There are two types of statistical methods, one of which can be used for summarizing a collection of data; this is known as descriptive statistics. This is practical within research when interpreting the experimental results. When one describes continuous data, the use of calculating the mean and standard deviation are used as numerical descriptors. However in categorical data, frequency and percentage are more useful. The second type of statistical method is known as inferential statistics. To infer means to reach a conclusion; similarly this method draws conclusions from data that are affected by random variation. These propositions are usually based on observed patterns with the sample data. By the scientific method, the propositions are investigated further and the conclusions are tested to support any hypotheses or observations. Such scrutinous testing is tested by correlation and modeling relationships with the data. Inference is vital for forecasting, predicting and estimating unobserved values in the sample population. The combination of the two aforementioned methods comprises applied statistics. By contrast, theoretical statistics uses logical reasoning to approach statistical inference. Theoretical statistics is also a branch of mathematical statistics, which does not only manipulate the probability distribution necessary for obtaining results related to inference, but has various aspects related to computational statistics.
6 6 PRINCIPAL COMPONENT ANALYSIS In statistics, the variance is an indicator of how spread out a set of numbers is. It is often used as a descriptor in probability distribution, which illustrates how far the set lies from the expected value (the mean). It is typically expressed in the following equation, which is simply the standard deviation squared, where n is amount of columns and X i is the data of column X i and X-bar is the mean. Covariance, on the other hand is the measurement of how two random variables interact and change together. If the greater values one of the two variables are similar to the greater values of the other variable, the covariance will be positive because it shows that they have behave in a similar pattern. By contrast, if the greater variables of one the variables can only correspond to the small values of the other variables, they represent opposite behavior and therefore their covariance will result in a negative value. Therefore it is clear that the sign of the covariance shows the interaction between variables. This can be easily observed in the following equation and is similar to variance except that is now multiplied to another data column rather than itself, where Y i and Y-bar is the data and mean amount of the second variable respectively. Invented in 1901 by mathematician Karl Pearson, principal component analysis is a procedure used in mathematics that applies orthogonal transformation to change a set of observations of presumed correlated variables into a set of linearly uncorrelated variables known as principal components. The amount of principal components tends to be less than or equal to the amount of presumed variables. The transformation is based on having the largest possible variance for the first principal component; this means that it accounts for that much variability within the data percentagewise. Each principal component succeeding the first has the highest variances under the constraint that it is orthogonal and thus uncorrelated with the preceding components. Principal component analysis is commonly used in exploratory data analysis and for making predictive models. One way that principal component analysis can be computed is by eigenvalue decomposition of a mean centered covariance matrix which is represented as followed with a 3x3 sample; the result should always be a symmetric matrix.
7 7 By spectral theorem, a symmetric matrix with real entries can be orthogonally diagonalized. For every matrix that satisfies this condition, there exists an orthogonal matrix Q with real entries such that D = Q T AQ is also a diagonal matrix. Therefore, every symmetric matrix is based on the choice of an orthonormal basis (orthonormal eigenvectors), which comprise the diagonal matrix. Every real symmetric matrix by extension is also Hermitian, and all eigenvalues are real values. Consequently, every complex symmetric matrix A can usually be diagonalized in the form of D = U T AU, where D represents the complex diagonal and U is a complex orthogonal (where U T U = I) of the eigenvectors of matrix A. Factor scores and loadings are the result of the computation, where factor scores are specific transformed variable values for a data point and factor loadings are the weight in which each originally presumed variable should be multiplied to get the component score. The operation of principal component analysis is helpful in illustrating the internal structure of the data with the depiction of a low dimension shadow image of the higher dimensional data space. This shadow is composed the first major components that capture most of the variance. Therefore the objective of principle component analysis is to find and reduce dimensionality of the data set and find the new meaningful variables. In order to use the principal component analysis method, the user needs a set of a data and then needs that data to be subtracted column by column by its column average to obtain the covariance matrix. After the covariance matrix has been obtained, find the eigenvalue and their normalized eigenvectors. Those eigenvectors will then form the factor loadings of each principal component, in order to calculate how much data it accounts for, the variance percent can be calculated by dividing each corresponding eigenvalue by the sum total multiplied by 100% for each principal component. The final step is to interpret the data depending on the experiment and the sample population. PC ANALYSIS OF A SUR VEY This survey was administered to psychology students at Vanier College. A total of 41 students were questioned in order to analyze their motivation in taking a statistic class. All of the students were of their second year at Vanier College and were asked to answer twelve questions by a scale from one to 4 (one being the most agreeing). The three types of motivations that were analyzed in the survey were each represented by four questions.
8 8 Questions: 1. Because without this statistics course it would be harder to get into university programs that lead to desirable jobs. 2. Because learning statistics is fun. 3. To prove myself that I am capable of succeeding in a statistics course. 4. Because succeeding in this course will help me to be accepted in excellent universities. 5. Because I enjoy learning new things about statistics. 6. Because I feel good about myself after I succeed in this course. 7. Because good grades in a statistics course are essential for admission to the best programs at University. 8. Because I enjoy learning statistical facts. 9. To prove myself that I am an intelligent person. 10. Because people who have good knowledge of statistics get ahead of others. 11. Because I will learn interesting things in a course in statistics. 12. Because I would be embarrassed if I cannot succeed in course about statistics. Meanings: The questions number 1, 4, 7 and 10 are related to extrinsic motivation, the regulation through identity, since it refers to a reward of wealth and intelligence that would put the person ahead of others (Having a better job due to a higher knowledge brings bigger salary and great approbation and desire by others). The questions number 2, 5, 8 and 11 are related to intrinsic motivation since it relies on personal satisfaction and pleasure. No external sources are involved, only a simple joy (The person is doing it because it is fun and interesting, there are no other reasons). The questions 3, 6, 9 and 12 are related to extrinsic motivation, the introjected motivation, because it refers to a personal goal, an achievement to be done in order to be happy or proud. It also involves the opinion of others (To prove yourself that you can do something and not being embarrassed if you fail is related to a goal and the opinion that others could have).
9 9 DATA AND ANALYSIS OF THE COVARIANCE MATRIX Table 1: Eigenvalues: Eigenvalue Eigenvalue Eigenvalue Eigenvalue Eigenvalue Eigenvalue Eigenvalue Eigenvalue Eigenvalue Eigenvalue Eigenvalue Eigenvalue Eigenvalue total Table 2: Eigenvectors: Eigenvector 1 {0.343;-0.330;-0.305;0.237;-0.366;-0.202;0.416;-0.197;-0.157;-0.104;-0.411;-0.183} Eigenvector 2 {-0.328;-0.062;-0.174;-0.406;-0.139;-0.396;-0.332;-0.158;-0.388;-0.244;-0.034;-0.413} Eigenvector 3 {-0.031; 0.075; ; 0.135; 0.006; 0.416; ; 0.277; ; ; 0.057;-0.820} Eigenvector 4 {0.371; 0.103; ; 0.126; 0.340; ;-0.007; 0.248; ; 0.239; 0.365; 0.218} Eigenvector 5 {-0.180; 0.130; ; 0.483; 0.333; ; 0.004; ; 0.055; ; 0.287; 0.167} Eigenvector 6 {0.385; ; 0.231; ; ; 0.360; ; ; ; ; 0.309; 0.112} Eigenvector 7 {0.058; 0.351; ; ; ; 0.404; 0.031; ; ; 0.111; 0.356; 0.006}
10 10 Eigenvector 8 {0.073; 0.618; 0.144; 0.348; 0.137; ; ; ; ; 0.157; ; } Eigenvector 9 {0.073; 0.618; 0.144; 0.348; 0.137; ; ; ; ; 0.157; ; } Eigenvector 10 {0.229; ; ; 0.110; ; 0.207; ; ; 0.161; 0.402; 0.219; } Eigenvector 11 {0.338; 0.073; 0.103; ; ; ; 0.040; 0.442; 0.530; ; ; } Eigenvector 12 {-0.012; ; ; ; 0.313; ; 0.154; 0.568; 0.186; ; ; 0.207} Table 3: Percentage of indexes: Factor (Index) Percentage of Variance Z % Z % Z % Z % Z % Z % Z % Z % Z % Z % Z % Z % The first four indexes account for a higher percentage of the variance than the other ones. This means that they contain more information about what is being tested than the others (which could be said insignificant). The first four together represent a total percentage data of 74.63%. Sample Calculations: Percentage Data: Z 1 = (2.144/7.293) x 100% = 29.40%
11 11 Percentage Data: Z 2 = (1.703/7.293) x 100% = 23.35% Percentage Data: Z 3 = (0.872/7.293) x 100% = 11.97% Percentage Data: Z 4 = (0.723/7.293) x 100% = 9.91% Equations: Where: Q = Extrinsic Motivation through Identity Regulation Q = Intrinsic Motivation Q = Extrinsic Motivation through Introjected Regulation Z 1 = (0.343)Q 1 + (-0.330)Q 2 + (-0.305)Q 3 + (0.237)Q 4 + (-0.366)Q 5 + (-0.202)Q 6 + (0.416)Q 7 + (-0.197)Q 8 + ( )Q 9 + (-0.104)Q 10 + (-0.411)Q 11 + (-0.183)Q 12 Z 2 = (-0328)Q 1 + (-0.062)Q 2 + (-0.174)Q 3 + (-0.406)Q 4 + (-0.139)Q 5 + (-0.396)Q 6 + (-0.332)Q 7 + (-0.158) Q 8 + (-0.388)Q 9 + (-0.244)Q 10 + (-0.034)Q 11 + (-0.413)Q 12 Z 3 = (-0.031)Q 1 + (0.075)Q 2 + (-0.012)Q 3 + (0.135)Q 4 + (0.006)Q 5 + (0.416)Q 6 + (-0.007)Q 7 + (0.277)Q 8 + ( )Q 9 + (-0.187)Q 10 + (0.057)Q 11 + (-0.820)Q 12 Z 4 = (0.371)Q 1 + (0.103)Q 2 + (-0.237)Q 3 + (0.126)Q 4 + (0.340)Q 5 + (-0.410)Q 6 + (-0.007)Q 7 + (-0.248)Q 8 + ( )Q 9 + (0.239)Q 10 + (0.365)Q 11 + (0.218)Q 12 For the purpose of the analysis, only the first index was taken to calculate the type and degree of motivation of the tested students. The first one was chosen because the coefficients representing the difference between the three types of motivations were more significant than in the other equations. Indeed, it is observable that for three quarter of the questions (leaving only the last three questions to be mixed up), the questions representing Identity Regulation are of positives numbers, that the Intrinsic Motivation are of negative numbers but of higher value than the Introjected Regulation. Therefore, it makes possible the analysis and relations between the type of motivations to each students. By replacing the number of a student s answer to the respective questions, it would give the value of the index (Z 1 ). If the value is positive, it means that the student had extrinsic motivation through identity regulation and if it is negative, a bigger number would mean intrinsic motivation and a smaller number would means extrinsic motivation through introjected regulation. However, it could be hard to determine the type of motivation if the numbers are not so different one from another. The other indexes were not taken since their coefficients did not really make sense and that they were not easy to analyse. For example, Z 2 only had negative numbers for coefficient and the values did not even respect an order (identity was not the highest everywhere, intrinsic the middle and introjected the smallest ) Similar mixes happened with the other indexes which would have made almost impossible a meaningful analysis.
12 12 COMPARISON Table 4: Students and their index value: Students Index value (Z 1 ) This table shows four students that were differently motivated. Obviously, regarding the positive value of the index, student #12 was extrinsic motivated through identity regulation, meaning that he was enrolled in the class because of a reward at the end of the class that would put him/her in a good situation (advantaged situation), a possibility of wealth and adoration by others. Also, it is pretty obvious that student #26 was intrinsic motivated due to the high negative value of the index. Therefore, this student was enrolled in the class before he/she liked it, found it interesting and amusing. Looking at students #6 & #39, they both have a negative value that is relatively small compared to student #26 which means that they were extrinsic motivated through introjected regulation. They were in the class for a personal achievement and recognition by others. However, student #27 also has a negative value, but the value is pretty much between and Therefore, it is difficult to determine whether the person was more intrinsic motivated or introjected. Moreover, calculating the index values of all the students showed that almost all the students represented an extrinsic motivation through introjected regulation or an intrinsic motivation due to the negative values. It is pretty odd that on 41 students, more than three quarter of them were all motivated between two ways. The odd results could have been due to the coefficient of question 10 that was also negative, which would not have given a possibility of more positive numbers.
13 13 CONCLUSION The collected data and the index equations showed that the survey did not really work in finding three distinct types of motivation of the students. It was seen with the first index, that represented 24.90% of the total information, but that the last coefficients did not correspond to the rest of the equation (identity regulation was supposed to have positive coefficient but was negative for the last one). Moreover, it was seen that even if the second index had 23.35% of the information, the equation could not lead to an effective calculation of the motivation of the students since the coefficient did not represented a significant difference between each other to be comparable. In addition, with the calculation of the index value for each student, it was seen that almost all of them were motivated between two ways; were motivated by the extrinsic motivation through introjected regulation or the intrinsic motivation. It was hard to tell which type exactly the students were motivated by due to the relatively close negative values obtained. Indeed, even by looking at the equations, it can be seen that there was no much of difference in the coefficient of the intrinsic and introjected motivation. This caused a mixture of data and a great confusion for the determination of the type of motivation. This fact is of concerned since it is rare, even almost impossible, for 41 students to be enrolled in a class for the same reason. Also, the questions asked might have been confusing for the students in the way that they did not really separated the difference between the tested motivations. This might be the reason why many students had 2 and 3 as answers to the questions; it represents the medium of the question (approximately corresponding to the question). The survey was discriminating enough to really make a concrete distinction between the types of motivations.
14 14 Bibliography Deci, E. L., & Ryan, R. M. (1985). Intrinsic motivation and self-determination in human behavior. New York: Plenum. Deci, E. L., & Ryan, R. M. (2000). The "what" and "why" of goal pursuits: Human needs and the self-determination of behavior. Psychological Inquiry, 11, Ryan, R. M., & Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. American Psychologist, 55, Website: Self-Determination Theory : consulted : & Website Wikipedia : consulted & Website Wikipedia : consulted Website Principal Component Analysis : consulted Website Isogenic statistical_analysis.html consulted Website consulted Website Wikipedia consulted
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