LITERATURE REVIEW: LOGISTICS AND LINEAR METHODS
Overview
Dystonia is a condition that affects the brain and the nervous condition of the patient (Jankovic 2005, 45). It causes some movement disorders whereby the patients’ muscles contracts without the patient ability to control it. This contraction of the muscles results in twisting of the body part which has been affected causing the patient to have an abnormal posture or repetitive movements. Women are more likely to get this condition as compared to men and it affects approximately one percent of the total population.
Symptoms of Dystonia
The symptoms can be mild or very severe and it can affect various body parts though the condition advances in stages. The initial symptoms are inclusive of cramping of the foot, uncontrollable eye blinking, leg dragging and speech difficulties (Jankovic, 2009, 65). This usually causes the patient to feel exhausted most of the time though stress and fatigue may worsen the situation.
The specific causes for dystonia cases are yet to be established though researchers in the medical field propose that it could be due to damage of the basal ganglia, which is responsible for nerve cells communication (Rittenhouse et al, 2008, 40). The condition is at time inherited or it can be acquired when the basal ganglia is damaged. The damage could have resulted from; brain trauma, tumor, stroke, oxygen deprivation among others (Smith, 2006, 56).
Phenomenology
Dystonia is a syndrome of a condition and it is not a disease. It forms part of one of the most difficult movement disorders that disorders that one can hardly recognize. This is because the condition does not always express itself in very conspicuous symptoms and this makes it to require a special attention and reliable equipment and specialists during the diagnosis (Walkup, 2006, 4). Dystonia is also majorly confused with a number of diseases and that makes it quite hard for the medic practitioners and the patient to receive an elaborate treatment when it is not detected at an early stage (Thrane et al, 2005, 300). The most common diseases that are mostly confused in this case includes tremor, chorea and even a condition called myoclonus. This confusion also comes because of varied ways in which it expresses itself and the very minimal distinguishing features form the rest of other disorders that is has. This is also because of the variability of the sped of the contractions. The direction of the contraction is almost always very consistent which is also characterized by the continuous involvement of the same or similar muscle group (Talpová et al, 2014, 348).
Classification of Dystonia
Usually dystonia classification is based on the affected body part. Dystonia which affects the general body parts is known as general dystonia. Then there is the focal dystonia which affects some specific parts of the body. Thirdly there is the multifocal dystonia which affects at least two body parts that are unrelated. The segmental dystonia affects adjacent body parts. Lastly there is the Hemidystonia usually affecting the hands and leg on the same body side (Conference on dystonia & Fahn, 2006, 33). The classifications and investigations of dystonia majorly relies on the genetic and the etiological criteria. This implies that the investigations can be traced from the family tree and history, the and anatomical distribution of the condition In the striato-cortical circuits , there are always physiological abnormalities that are common in dystonia although their pathophysiology always still remain not to be clear. This paper also gives the classification, phenomenology, etiologies, the investigation, treatment and the general pathophysiology of dystonia.
Treatment of Dystonia
The cure for dystonia is yet to be determined and therefore efforts of treatment are meant to lessen the symptoms. The major goal is on the improvement of the patient’s quality of life. In order to establish a good treatment plan patients’ needs to be involved by the medical practitioners (Shang et al, 2000, 290).
Usually the first step in dystonia treatment is establishing the cause of the condition after which there are different therapies that are administered to the patient (Conference on dystonia & Fahn, 2006, 33). Complementary therapies are help when considered in addition to the medications and the surgery that the patient undergoes. The overall treatment plan should be holistic involving both the emotional wellbeing such as mental health, the body comprising of physical medicine and the spiritual well being which is comprised of support given to the patient (Stacy, 2007, 68). It is therefore imperative that dystonia treatment is both an art and a science and none should be over emphasized.
Epidemiology
Dystonia is characterized with so many etiologies and the prevalence of the condition is unknown. Dystonia is the most prevalent movement disorder after the essential Tremor (ET) and after the Parkinson`s disease (PD). the overall determined value of the prevalence for the general dystonia is 3.4 per 100,000 and it is 29.5 per 100,000 for the focal dystonia.in dystonia, there exists ethnic differences prevalence where at the childhood and at adolescent-onset stages primary dystonia which is becoming more common in Jews of the Eastern Europe (Arana, 2014, 3480).
Linear regression
Overview
Linear regression attempts to find how two variables relate by use of a linear equation in the observed data (Montgomery 2011). With the two variables selected one is assumed to be the dependent variable whereas the other becomes the explanatory variable. For example nutritionist usually uses this concept to determine the Basal Body Index (BMI) through using the relationship between the body weight and the height.
Linear model should only be applied when there is a relationship between the variables at hand (George, 2008, 56). Important to note however is that the variables do not necessarily cause the other, there should be some level of association between the variables. Chatterjee and Hadi (2006, 56) asserts that In order to determine the strength of relationships between variables scatter plot can be of relevance. Correlation coefficient is a significant numerical measure of relationship between two variables. The values of the correlation coefficient normally fall between -1 and 1 which indicates the strength of the association of the given data for variables being used (Ge & Whitmore, 2011, 90).
The equation of a linear regression line is Y= b0+ Xb1, where X is the explanatory variable whereas Y is the dependent variable, b1 refers to the line’s slope and to b0 the intercept (Adwere-Boamah, 2011, 5).
Problem Statement
A hospital had admissions of four patients all of which were found to be having dystonia condition. The ages of the patients admitted was recorded. Over the years the medical practitioners had hypothesized that there is a relationship between the patient’s age and the heart beat rate which is significant in determining the level of fatigue of the patient after doing some mid morning exercise. One patient had the age of 80 and was used as the reference point.
Regression Equation
In order to determine the regression equation the results of the aptitude tests and the score for results of accuracy in research analysis are required. Presentation of the scores in a table was as follows.
Using the formulae for finding the linear equation, Y= b0+ Xb1 conducting an analysis will demand that the values of b0 and b1 be determined.
The regression eqution Y is therefore = 26.768+ 0.644x
Using the Linear Equation
Using regression equation is very easy once the calculation of the equation has been determined (Weisberg, 2005, 45). The next step is selecting a random value for the independent variable (x), estimated value for Y is determined after computing. In the case study for patients with dystonia the independent variable becomes the patients age where as the dependent variable is the patient heart beat rate after the mid morning exercise. The expected heart beat rate for the patient who is 80 years after the mid morning exercise will hence be:
Y = 26.768 + 0.644x = 26.768 + 0.644(80) = 26.768 + 51.52 = 78.288
Outliers in Linear Regression
Outliers refer to scores in the linear regression that have values of the independent variable that differs a lot from the range of values that was used in creating the equation. This values have got have high residual values (Montgomery, 2011, 35). The values are usually referred to as extrapolation and have the tendency of producing unrealistic estimates. In this example the scores for research analysis accuracy ranged within 60 to 95 and hence values used for estimates should lie within this range to avoid issues of extrapolation.
Coefficient of Determination
Coefficient of determination is usually used in determining how effectiveness of the equation while doing data calculations. There is a formula used for computing the coefficient of determination as follows; formula.
R2 = { ( 1 / N ) * Σ [ (xi -xm) * (yi - ym) ] / (σx * σy ) }2
N is the observation number applied in fitting the model, Σ is the symbol for summation of the values, xi is the value of x for i observation. xm represents the value of x mean xm, yi is the value of y for i observation. ym represents the value of y mean ym, σx is the standard deviation for x and σy is the standard deviation for y. In this case example the coefficient of correlation is determined by:
R2 = { ( 1 / N ) * Σ [ (xi - xm) * (yi - ym) ] / (σx * σy ) }2 R2 = [ ( 1/5 ) * 470 / ( 12.083 * 11.225 ) ]2 = ( 94 / 135.632 )2 = ( 0.693 )2 = 0.48
The score of 0.48 is equivalent to a percentage of 48 which is actually a good fit for this data in that it can predict statisticians score on research analysis.
Linear regression is an important concept for all researchers and it is one of the simplest tools for determining relationships between variables.
Logistic Regression
Overview
Statistical models are divided into different categories and logistic variation a part of generalized linear models. This wide class of models is inclusive of ANOVA, ordinary regression, ANCOVA (which is a miultivariate statistic) and loglinear regression (Agresti, 2006, 325). Logistic regression is applicable in predicting discrete outcomes like membership of a group, from variables that may be discrete, dichotomous, and continuous or a mixture of this all (Soureshjani & Kimiagari, 2013, 1420).
Usually the dependent variable is dichotomous for instance failure or success (Menard, 1995, 34). In essence since the dependent variable is dichotomous it takes a value of one with chances of success , or value of 0 with a chance of failure 1-. This is a kind of variable referred to as Bernoulli or otherwise binary variable. Chen & Popovich (2006, 76) asserts that multinomial, polychotomous or polytomous is a name given to a form of logistic regression in which the dependent variable is of more than two cases.
The independent variable in logistic regression takes any form. This is because logistic regression gives no assumption on the distribution of the independent variables (Qiu-Hao et al, 2007, 50). The relationship between the dependent variable and the independent variable is determined through logistic regression function (logit transformation of :
Where = the constant of the equation, = the coefficient of the predictor variables.
Problem Statement
The health practitioners working with patients with the dystonia condition wish to analyze the life expectancy of the patients from the time they contacted the condition. Physical exercise have been observed and found useful in health improvement. The results for 64 dystonia patients are presented in the table below:
X represents the years in which the patient live after contracting the condition
Y represents whether the patient used to exercise after contracting the disease (‘no’ coded as ‘0’ and entered in column (2); ‘yes’ coded as "1" and entered in column (3))
The table also shows:
5. The observed probability of Y=1 for each x, calculated as ratios of numbers of y=1 instances to the totals of y instances in the same level.
6. The x odds, calculated as ratio of y=1 entries to the numbers of y=o in each level.
7. The logarithm of the odds for specific levels of x, referred to as Log Odds.
Using the formula or from plotting a graph the intercept y is found to be 17.2086 where as the regression line X is equal to 0. The slope or the constant is 0.5934, when exponeted it gives a value of 0.5934= 1.81.
In this example the predicted X=31 log = -17.2086+ (0.5934*31) = 1.1868
The corresponding predicted odds are
Odds = exp(Log(odds)) =exp(1.1868) = 3.2766
The corresponding predicted probability is;
Probability+ odds/ (1+odds) = 3.2766) =0.7662, this results indicates that there is a significant relation between exercising and having a prolonged life.
Research Contribution
Many researchers have stormed into the use of logistic and linear regressions in analysis of factors causing dystonia. Most of these analysis have been highly concentrated on the relationship between the causation factors and the diseases. However, there is little done in realizing the powerful nature of logistic and linear regressions in detecting how the presence of the disease can affect the life of the individuals in the long run (Akinci et al, 2007, 550). This research is significant as it will contribute the patients and medical practitioners affected by dystonia in knowing some of the correlations between the condition and other variables such as health, the age, social support given to the patient and enhancement of life quality. The research also contributes in showing how regression analysis can be applied in the medical world in order to come up with solutions to various challenges that medical practitioners and their patients go through (Okun, 2006, 78).
Many people who have relatives and friend suffering from the condition of dystonia are likely to benefit from this paper. This is because the paper gives the important of social support system to the patient with dystonia. When patients are given support they are more likely to have a sense of significance and to take better care of themselves so as to increase their life quality (Wilson 2002, 56).
The knowledge applied in this paper is useful for all scholars and especially those in the faculty of statistics and mathematics. The example given both for the linear regression and the logistic regression are simple and easy to follow. The formulas have also been indicated and the relevant explanations given for the research so that it becomes very relevant for future scholars who would wish to do researches related to this topic. When carrying out any scientific research it is important that the researchers are able to establish reasons as to why they are doing the research. Research should be relevant and provide helpful information that can enhance development in the specific field that the research is done (Wilson 2002, 56).
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