SNAPSHOT.01 EZESTAT1 旼Page 11컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿟ypes of variables (1) � � � 쿔n statistics, a distinction is made between : � � � 쿎ATEGORICAL variables � � � 쿟hese are essentially coding numbers and are often used to separate � 쿶tems in a list into various categories. � � � 쿑or example, 1=Female 2=Male is one common example of a categorical � � variable � � � � 1=Agree 2=Neutral 3=Disagree would be another example. � � � 쿎ONTINUOUS variables � � � 쿟hese are numbers that are 'real' ( as opposed to the artificiality of the � 쿻umbers often used in categorical variables) They are used for data that � 쿬an form a long continuous stream such as heights, weights or incomes. � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.02 EZESTAT1 旼Page 12컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿟ypes of variables (2) � � � � � 쿐ach of the variables are measured by different families of statistics � � � 쿎ATEGORICAL variables are measured by NON-PARAMETRIC statistics � � (such as contingency tables, chisquare) � � � 쿎ONTINUOUS variables are measured by PARAMETRIC Statistics � � (such as a mean, standard deviation or a t-test) � � � � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � T E S T Y O U R U N D E R S T A N D I N G � � � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � Would you measure a band of expenditure (e.g. � � � � �1-�10 �11-20 �31-�40 �41 or more � � � � by means of a categorical or a continuous variable ? � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � � � � � � [1] Continuous [2] Categorical � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.03 EZESTAT1 旼Page 12b컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � T E S T Y O U R U N D E R S T A N D I N G � � � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � Would you measure a band of expenditure (e.g. � � � � �1-�10 �11-�20 �31-�40 �41 or more � � � � by means of a categorical or a continuous variable ? � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � 쿧ou answered � 쿬ategorical � 퀇hich was CORRECT. � � � 쿌lthough incomes are being measured, they are placed into categories � 쿪nd would probably have a coding number associated with each one � 쿮.g. 1 � 1-�10 � � 2 �11-�20 � � 3. �21-�30 � 쿪nd so on. So this data is actually measured at the categorical level. � 쿧ou were not confused by the fact that the data appears to be continuous � 쿫ut is, in fact, categorical because it was placed into bands. � � � � [Pg Dn] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.04 EZESTAT1 旼Page 13컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎orrelation 1 � � � 쿔n the diagram below, we imagine that we are plotting the amount spent � 쿽n food and the income received for a sample of 12 families : � � � � � � � � x x Notice that : � � � x � � Food (=y) � x x Food (y) is the DEPENDENT variable� � � x Income (x) is the INDEPENDENT � � (dependent) � x x variable � � � x � � � x x As x increases, so does y � � 퀈 � � 읕컴컴컴컴컴컴컴컴컴컴컴� � � Income (=x) (independent) � � � 쿟his is an example of a HIGH POSITIVE correlation � � � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.05 EZESTAT1 旼Page 14컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎orrelation 2 � � � 쿥e are now plotting the amount of days recorded as 'illness' and income � 퀁eceived for a sample of 12 families : � � � � � � � � x Notice that : � � � x x � � Illness(=y) � x x As income (x) INCREASES, the number� � � x x of days of 'illness' (y) DECREASES � � (dependent) � x x � � � x x (i.e. Higher Income families do not� � � x have as much 'illness' as � � � Lower Income families ) � � 읕컴컴컴컴컴컴컴컴컴컴컴� � � Income (=x) (independent) � � � 쿟his is an example of a HIGH NEGATIVE correlation � � � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.06 EZESTAT1 旼Page 15컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎orrelation 3 � � � 쿥e are now plotting the 'Final Degree Average' obtained against the � �'A-level' points score for a sample of 12 students: � � � � � � � � x x Notice that : � � Final � x x � � Degree � There are as many 'good' degrees � � Average (=y)� x x associated with 'good' A-level � � (dependent) � x scores as there are 'poor' degrees � � � x x � � � x x (i.e. there does not appear to be � � � x any association between � � 읕컴컴컴컴컴컴컴컴컴컴컴� 'A'-level points' and � � 'A'-level point score (=x) 'Final Degree Average') � � � � � 쿟his is an example of NO association at all � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.07 EZESTAT1 旼Page 16컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎orrelation 4 � � � 쿌 correlation coefficient is measured in the range -1....0....+1 � � � 쿔t is usually designated like this : r= 0.9 or r=-0.85 � � � � � 쿌 value in the range 0 +0.3 } is regarded as a LOW positive � � 0 -0.3 } ( or negative) correlation � � � 쿌 value in the range +0.4 +0.6 } is regarded as a MODERATE positive � � -0.4 -0.6 } ( or negative) correlation � � � 쿌 value in the range +0.7 +1.0 } is regarded as a HIGH positive � � -0.7 -1.0 } ( or negative) correlation � � � 쿝emember that values that cluster around 0 represent NO ASSOCIATION � 쿪nd not a negative association. The closer values get to +1 ( or to -1), � 퀃he more they measure a HIGH positive ( or negative) correlation. � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.08 EZESTAT1 旼Page 17컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎orrelation 5 � � � � � 쿘entally divide the plot into 4 quadrants : � � � � � | If there is a clustering from bottom left to top � � � | right [ / ] this is a POSITIVE correlation. � � �------|---�-- � � � | If there is a clustering from top left to bottom � � � | right [ \ ] this is a NEGATIVE correlation. � � 읕컴컴컴컴컴컴 � � If the data appears to be equally split between � � all four quadrants, then this represents a � � pattern of NO ASSOCIATION or very low correlation. � � � 쿕ust because two variables are associated with each other, we cannot � 퀂ay that one is the cause of the other. � � � 쿝emember that there is a high correlation between storks and babies! � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.09 EZESTAT1 旼Page 18컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � T E S T Y O U R U N D E R S T A N D I N G � � � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � � � � � � x � � � � 퀈 x Do you think the � � � � � x correlation coefficient � � � � � x in this case is : � � � � � � � � � � x x 0.1 � � � � � x x -0.7 � � � � � x -0.4 � � � � 읕컴컴컴컴컴컴컴컴컴컴컴 � � � � � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � � � � � � � � � � � � [1] 0.1 [2] -0.7 [3] -0.4 � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.10 EZESTAT1 旼Page 18c컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � T E S T Y O U R U N D E R S T A N D I N G � � � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � � � � � � x � � � � 퀈 x Do you think the � � � � � x correlation coefficient � � � � � x in this case is : � � � � � � � � � � x x 0.1 � � � � � x x -0.7 � � � � � x -0.4 � � � � 읕컴컴컴컴컴컴컴컴컴컴컴 � � � � � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � 쿧ou answered -0.4 which is CORRECT. If you divide the plot (mentally) � 쿶nto four quadrants, you will a see more of a clustering from top left � 퀃o bottom right which indicates a negative relationship. However, this � 쿶s not very marked and therefore -0.4 is an accurate correlation � 쿬oefficient for this plot. � � � � [Pg Dn] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.11 EZESTAT1 旼Page 19컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎orrelation and Causation (1) � � � � � x x The plot shows a very high correlation between � � � x x EDUCATION (years spent in full-time post 18) and � � � x xx x INCOME. � � � x x x � � � x x x Before we can assume that many years spent in post- � � � x x x school education CAUSES high income, we must examine� � � x x the impact of other factors e.g. social class which � � 읕컴컴컴컴컴컴컴� might intervene and EXPLAIN the relationship. � � � 쿔n this particular case, if we divided the group into : � 쿛rofessional Parents v. Non-Professional parents � 퀇e might discover that � � � 쿛rofessional parents - children ARE highly educated and have HIGH incomes � 쿙on-professional - children ARE NOT highly educated and have LOW incomes� � parents � � � 쿪nd therefore SOCIAL CLASS explains high incomes and NOT education 'per se' � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.12 EZESTAT1 旼Page 20컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎orrelation and Causation (2) � � � 쿟herefore, just because two factors are correlated, we cannnot say that one � 쿬auses the other - nor can we say that one does NOT cause the other. � � � 쿎orrelation means that where we find one factor, we also find the other � 쿑OR WHATEVER REASON. � � � 쿎orrelation does not imply causation! � � � � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � T E S T Y O U R U N D E R S T A N D I N G � � � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � If you found that there was a high correlation between � � � � smoking and lung cancer, can we say that smoking � � � � causes lung cancer ? � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � � � � � � � � [1] Yes [2] No � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.13 EZESTAT1 旼Page 20b컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � T E S T Y O U R U N D E R S T A N D I N G � � � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � If you found that there was a high correlation between � � � � smoking and lung cancer, can we say that smoking � � � � causes lung cancer ? � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � 쿧ou answered � 쿙O � 퀇hich is the CORRECT answer. � � � 쿌lthough medical evidence does indeed suggests that smoking is closely � 퀁elated to lung cancer, we cannot PROVE that smoking causes lung cancer � 쿷ust because the two factors are closely correlated. � � � 쿥e would have to find an ACTIVE link (i.e. carcinogen) before we can say � 퀃hat smoking causes lung cancer. It is logically possible (although not � 쿹ikely) that a third factor (such as a genetic factor) predisposes people � 퀃owards smoking AND also makes them more prone to lung cancer. � � � � � � [Pg Dn] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.14 EZESTAT1 旼Page 21컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎urvilinear data � � � � � x x Very often in the social and biological sciences, � � � x the data does not follow a straight line pattern � � � x and is said to be CURVILINEAR. � � � x � � � x This means that it follows a natural growth � � � x curve ( like an 'S' ) in which there is: � � � x x � � 읕컴컴컴컴컴컴컴� � � � � � An initial period of slow growth (bottom of the curve) � � � A rapid take-off of sustained growth eventually followed by � � � a tailing off in rate of growth (top of the curve) � � � 쿐.g. Although all individuals continue to grow in height/gain weight � � after the age of 18, they do not do so at the same rate that � � characterised earlier years of adolescence i.e. from ages 14-18. � � � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.15 EZESTAT1 旼Page 22컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎alculating Spearman's coefficient of rank correlation � � � 쿔n the data set, Household Income (x) and Food expenditure (y),each column � 쿽f data is RANKED from lowest to highest, the difference between the ranks � 쿶s squared and finally we calculate the sum of the ranked differences : � � � � Income Food Income Food Difference Difference� � � (x) (y) (Rank) (Rank) � � 17.0 4.0 3 1 2 4 � � 16.0 4.8 2 2 0 0 � � 12.0 6.2 1 3 2 4 � � 24.0 7.4 4.5 4 0.5 0.25 � � 30.0 9.0 7 7 0 0 � � 28.0 8.8 6 6 0 0 � � 24.0 8.2 4.5 5 0.5 0.25 � � � 쿟he sum of the differences squared (�d�) is 8.5. When we have 'tied � 퀁anks', (e.g. 2 cases of 24) then we 'split' the appropriate ranks between � 퀃hem (i.e. 4 and 5 in this case, giving 4.5) and the next position is 6. � � � � � � [Pg Dn] [Pg Dn] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.16 EZESTAT1 旼Page 23컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿌pplication of the Spearman formula of Rank correlation � � � � 6�d� Note : the 6 is a constant � 쿑ormula : 1 - 컴컴컴컴� i.e. it is ALWAYS � � n(n2-1) a 6 � � � � � � 6(8.5) Note : n is the number of � � = 1 - 컴컴컴컴 data PAIRS � � (7)(49-1) (7 in this case) � � � � � � 51 Note : Work out the whole of � � = 1 - 컴컴컴컴� the right hand side � � 336 and THEN subtract from 1 � � � � = 1 - 0.1518 ( to 4 decimal places) � � � � = 0.8482 Note : must be in the range � � -1 ... 0 ... +1 � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.17 EZESTAT1 旼Page 24컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿛earson Product Moment Correlation Coefficient � � � 쿔n the data set, Household Income (x) and Food Expenditure (y), we have � 퀃o calculate SIX values before we start to apply the formula : � � � � �x �x� �xy (multiply x by y and then add them all together) � � �y �y� n (the number of data PAIRS) � � � � Income (x) Food (y) � � 17.0 4.0 �x (sum of x) = 151.00 � � 16.0 4.8 �y (sum of y) = 48.40 � � 12.0 6.2 �x� (sum of x's squared) = 3525.00 � � 24.0 7.4 �y� (sum of y's squared) = 357.92 � � 30.0 9.0 �xy (sum of xy's) = 1110.00 � � 28.0 8.8 n (number of data pairs) = 7.00 � � 24.0 8.2 � � � 쿟his will take THREE passes through your calculator (clear it each time): � � (1) for the �x,�x� (2) for the �y,�y� (3) for the �xy � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.18 EZESTAT1 旼Page 25컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿌pplication of the Pearson Product Moment Correlation formula : � 념x = 151.0 �x� = 3525.0 �y = 48.4 �y� = 357.92 �xy = 1110 n=7 � � � � Note: Think of the formula as: � � � � n�xy - �x�y Top Line � � r = 컴컴컴컴컴컴컴컴컴컴컴컴컴컴 컴컴컴컴컴컴컴컴컴컴컴컴컴컴 � � �(n�x�-(�x)�).(n�y�-(�y)�) �(Bottom Left).(Bottom Right) � � � � � � = (7)(1110) - (151)(48.4) Note : do multiplication � � 컴컴컴컴컴컴컴컴컴컴컴컴컴� before subtractions� � �((7)(3525-(151)�)).((7)(357.92-(48.4)�)) both top and bottom� � � � = 461.6 Note : Multiply both the � � 컴컴컴컴� bottom brackets and� � �(1874)(162.88) then take the � � � � � = 0.8355 Note : The Answer ! � � Must be in the range -1 ... 0 ... +1 � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.19 EZESTAT1 旼Page 26컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎omparison of Pearson and Spearman Correlation Coefficients � � � � �袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴藁袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴敲 � � � SPEARMAN coefficient � PEARSON coefficient � � � 훤袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴姦袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴� � � Advantages Advantages � � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴쩡컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � Generally easy and quick � More accurate because it uses � � � � to calculate � the exact values in each pair � � � � � of values � � � � � � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴좔컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � Disadvantages Disadvantages � � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴쩡컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � Less accurate because it uses � The calculations can be long � � � � data that has been made less � and error prone � � � � exact (i.e. degraded) by being � � � � � put into ranks � � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴좔컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.20 EZESTAT1 旼Page 27컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿝egression (1) � � Plot of Food versus Income � � � . Actual data plots : . � � � . * Regression line : * � � � * � � � � . * � � � Food (=y) � * �<- SLOPE which is the ratio of � � � * .컴컴컴컴컴컴켸 the vertical to the � � � . * horizontal difference ( =b ) � � � * . � � �* . A REGRESSION LINE is an � � � � equation which is used to � � INTERCEPT� � PREDICT future values of � � ( =a ) � � y given any particular x � � � 聃컴컴컴컴컴컴컴컴컴컴컴컴컴 � � Income (=x) General form : y = a + bx � � where a=intercept, b=slope � � � 쿟he regression equation for the graph above is : y = 1.6009 + 0.2463x � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.21 EZESTAT1 旼Page 28컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � Income (x) Food (y) � � 17.0 4.0 �x (sum of x) = 151.00 � � 16.0 4.8 �y (sum of y) = 48.40 � � 12.0 6.2 �x� (sum of x's squared) = 3525.00 � � 24.0 7.4 � 30.0 9.0 �xy (sum of xy's) = 1110.00 � � 28.0 8.8 n (number of data pairs) = 7.00 � � 24.0 8.2 � � � 쿧ou will have this data from having calculated the correlation coefficient. � � � 쿍efore you start to calculate the regression equation, determine which � 퀆alue is to be the x (independent) and which the y (dependent) value.. � � _ � � FORMULAE [ often, you will see �x/n expressed as x � � and similarly for �y/n ] � � � � �xy - (�x/n).�y � � b = 컴컴컴컴컴컴컴컴 a = (�y/n) - b.(�x/n) � � �x� - (�x/n).�x � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.22 EZESTAT1 旼Page 29컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎alculation of the Regression equation � � � 념xy = 1110 �x = 151 �y = 48.4 �x� = 3525 n = 7 � � � � �xy - (�x/n).�y � � b = 컴컴컴컴컴컴컴� � � �x� - (�x/n).�x � � � � = 1110 - (151/7)(48.4) � � 컴컴컴컴컴컴컴컴컴컴컴� � � 3525 - (151/7)(151) � � � � = 65.9429 旼> a = (�y/n) - b.(�x/n) � � 컴컴컴� � � � 267.7143 � = (48.4/7) - (0.24632)(151/7) � � � � � = 0.24632 컴컴컴켸 = 1.6008 � � � 쿟he regression equation is : y = 1.6008+0.24632x � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.23 EZESTAT1 旼Page 30컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿢sing the regression equation to predict values � � � 쿟he whole purpose of a regression equation is to use past data to interpet � 쿯uture trends. � � � 쿔f we have the regression equation � � � � y = 1.6008 + 0.24632x (where x=Income, y=Food) � � � 퀃hen we can predict that a family with an income of �100 a week will spend � � � � y = 1.6008 + (0.24632)(100) = 1.6012 + 24.63 = �26.23 a week on food � � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � T E S T Y O U R U N D E R S T A N D I N G � � � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � In the case of correlation and regression, does it matter � � � � which is x (independent) and which is y (dependent) ? � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � [1] It does not matter which is which � � [2] It matters for correlation but not for regression � � [3] It matters for regression but not for correlation � � [4] It does matter in the case of both correlation and regression � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.24 EZESTAT1 旼Page 30c컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � T E S T Y O U R U N D E R S T A N D I N G � � � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � In the case of correlation and regression, does it matter � � � � which is x (independent) and which is y (dependent) ? � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � 쿧ou answered � � � 쿶t matters for regression but not for correlation � � � 퀇hich is the CORRECT answer. You have remembered correctly that it � 쿶s important to get the 'correct' values for x and y in a regression. � � � 쿟he formula for correlation is symmetrical, so it does not matter which � 쿶s x and which is y. But it DOES matter for regression, as the shape of � 퀁egression line of 'x upon y' is different to the shape of the line of � �'y upon x' � � � � � � � � [Pg Dn] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.01 EZESTAT2 旼Page 29컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎ontingency tables � � � 쿝emember that we can collect data at the categorical (nominal) level - each � 쿻umber is essentially a CODING number for categories in the data. � � � 쿞EX 1 = Female 2 = Male [ Data 1 2 1 2 1 2 1 2 1 2 ] � 쿏RIVER 1 = Driver 2 = Non-driver [ Data 1 1 2 1 1 2 1 1 2 1 ] � � � 쿥hen we collect data for two variables, each split into several categories, � 퀃hen it is possible to form a CONTINGENCY table. In a contingency table, � 쿮ach cell hold the totals of the categories formed by the axes. � � DRIVER-> Driver Non-Dr � � 1 2 We have 3 1,1 [ Female drivers ] � � 旼컴컴쩡컴컴� 2 1,2 [ Female non-drivers ] � � SEX F 1 � 3 � 2 � 7 4 2,1 [ Male drivers ] � � 쳐컴컴탠컴컴� 1 2,2 [ Male non-drivers ] � � M 2 � 4 � 1 � 3 � � 읕컴컴좔컴컴� and we would call this a '2 x 2' table � � 5 5 10 � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.02 EZESTAT2 旼Page 30컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎omplex contingency tables � � Crosstabulation of SEX by DRIVER Holds driving licence � � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � DRIVER >쿧es No � ROW TOTAL � Complex contingency table � � � SEX � 1 2 � � ~~~~~~~~~~~~~~~~~~~~~~~~~ � � � 쳐컴컴컫컴컴컴� � This contingency table (taken � � � 1� 3 � 2 � 5 � from TURBOSTATS) contains in � � � [Row %] � 60.0 � 40.0 �50.0% � each cell : � � � [Col %] � 42.9 � 66.7 � � � � � [Tot %] � 30.0 � 20.0 � � - the cell COUNTS � � � 쳐컴컴컵컴컴컴� � - the ROW percentages � � � 2� 4 � 1 � 5 � - the COLUMN percentages � � � [Row %] � 80.0 � 20.0 �50.0% � - the TOTAL percentages � � � [Col %] � 57.1 � 33.3 � � � � � [Tot %] � 40.0 � 10.0 � � We also see the MARGINAL � � � 읕컴컴컨컴컴컴� � TOTALS and PERCENTAGES � � � TOTAL 7 3 10 � for each column and each � � � 70.0% 30.0% 100.0% � row in the table. � � � Valid cases = 10 Missing = 0 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.03 EZESTAT2 旼Page 31컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎hi-square table (1) � � Crosstabulation of SEX by DRIVER Holds driving licence � � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � DRIVER >쿧es No � ROW � Chi-square calculation (1) � � � SEX � 1 2 � TOTAL � ~~~~~~~~~~~~~~~~~~~~~~ � � � 쳐컴컴컫컴컴컴� � 'Exp' is the EXPECTED value in � � � 1� 3 � 2 � 5 � each cell. We calculate it by � � � [Exp ] � 3.5 � 1.5 �50.0% � using the ROW & COLUMN totals : � � � [Chis ] � 0.07� 0.17� � � � � 쳐컴컴컵컴컴컴� � 5/10 are Females whilst 7/10 � � � 2� 4 � 1 � 5 � are Drivers. � � � [Exp ] � 3.5 � 1.5 �50.0% � � � � [Chis ] � 0.07� 0.17� � We would EXPECT 7 * (5/10) =3.5 � � � 읕컴컴컨컴컴컴� � Female Drivers in the top left � � � TOTAL 7 3 10 � hand cell. [ Each other cell is � � � 70.0% 30.0% 100.0% � worked out by a similar logic ] � � � Valid cases = 10 Missing = 0 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � Total chi-square D.F. Significance Cells with E.F. < 5 � � 0.48 1 0.4902 4 of 4 ( 100.0% ) � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.04 EZESTAT2 旼Page 32컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎hi-square table (2) � � Crosstabulation of SEX by DRIVER Holds driving licence � � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � DRIVER >쿧es No � ROW � Chi-square calculation (2) � � � SEX � 1 2 � TOTAL � ~~~~~~~~~~~~~~~~~~~~~~ � � � 쳐컴컴컫컴컴컴� � The individual chi-square in each � � � 1� 3 � 2 � 5 � cell is worked with the formula: � � � [Exp ] � 3.5 � 1.5 �50.0% � 2 � � � [Chis ] � 0.07� 0.17� � (Observed - Expected) � � � 쳐컴컴컵컴컴컴� � 컴컴컴컴컴컴컴컴컴컴� � � � 2� 4 � 1 � 5 � Expected � � � [Exp ] � 3.5 � 1.5 �50.0% � e.g. � � � [Chis ] � 0.07� 0.17� � in top left-hand cell, chi-square � � � 읕컴컴컨컴컴컴� � = (0.5)*(0.5)/3.5 = 0.07 � � � TOTAL 7 3 10 � � � � 70.0% 30.0% 100.0% � The chi-squares are then TOTALLED � � � Valid cases = 10 Missing = 0 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � Total chi-square D.F. Significance Cells with E.F. < 5 � � 0.48 1 0.4902 4 of 4 ( 100.0% ) � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.05 EZESTAT2 旼Page 33컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎hi-square table (3) � � Crosstabulation of SEX by DRIVER Holds driving licence � � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � DRIVER >쿧es No � ROW � Chi-square calculation (3) � � � SEX � 1 2 � TOTAL � ~~~~~~~~~~~~~~~~~~~~~~ � � � 쳐컴컴컫컴컴컴� � The D.F. (Degrees of Freedom) � � � 1� 3 � 2 � 5 � figure is worked out by the � � � [Exp ] � 3.5 � 1.5 �50.0% � formula : (rows-1) * (columns-1) � � � [Chis ] � 0.07� 0.17� � � � � 쳐컴컴컵컴컴컴� � In a '2 x 2' table (as here) then � � � 2� 4 � 1 � 5 � DF = (2 - 1) * (2- 1) = 1 � � � [Exp ] � 3.5 � 1.5 �50.0% � � � � [Chis ] � 0.07� 0.17� � The Significance figure (p) needs � � � 읕컴컴컨컴컴컴� � to be < 0.05 to be significant � � � TOTAL 7 3 10 � (i.e. less than 1 chance in 20 � � � 70.0% 30.0% 100.0% � of getting this result by chance) � � � Valid cases = 10 Missing = 0 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � Total chi-square D.F. Significance Cells with E.F. < 5 � � 0.48 1 0.4902 4 of 4 ( 100.0% ) � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.06 EZESTAT2 旼Page 34컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎hi-square table (4) � � Crosstabulation of SEX by DRIVER Holds driving licence � � 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � DRIVER >쿧es No � ROW � Chi-square calculation (4) � � � SEX � 1 2 � TOTAL � ~~~~~~~~~~~~~~~~~~~~~~ � � � 쳐컴컴컫컴컴컴� � If the number of cells with an � � � 1� 3 � 2 � 5 � Expected Frequency (EF) < 5 are � � � [Exp ] � 3.5 � 1.5 �50.0% � less than 20% of the total number � � � [Chis ] � 0.07� 0.17� � of cells, then some of the � � � 쳐컴컴컵컴컴컴� � assumptions of the chisquare test � � � 2� 4 � 1 � 5 � are being violated. � � � [Exp ] � 3.5 � 1.5 �50.0% � � � � [Chis ] � 0.07� 0.17� � The solution lies in COLLAPSING � � � 읕컴컴컨컴컴컴� � some of the categories (if � � � TOTAL 7 3 10 � possible) to reduce the total � � � 70.0% 30.0% 100.0% � number of cells in the table. � � � Valid cases = 10 Missing = 0 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � Total chi-square D.F. Significance Cells with E.F. < 5 � � 0.48 1 0.4902 4 of 4 ( 100.0% ) � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.07 EZESTAT2 旼Page 38컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿝ecoding data � � � 쿥e may need to recode data to turn CONTINUOUS into CATEGORICAL data... � 쿔NCOME Y_CODE SEX � � 1000 1 1 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � 2000 1 2 � Here we have INCOME (a continuous variable) � � � 3000 1 1 � coded into Y_CODE (a categorical variable) � � � 4000 1 2 � We could now, for example, do a chi- square � � � 5000 2 1 � test on Y_CODE v SEX � � � 6000 2 2 � � � � 7000 2 1 � NB We do lose some of the 'richness' in � � � 8000 2 2 � the data when we code like this. � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � � � 쿥e can also COLLAPSE data, which is similar to recoding. If we have a � 퀆ariable CLASS measured on a six-point scale [1-6] then we could recode � �1-3 into a 1 [ Middle class ] � �4-6 into a 2 [ Working Class] � 쿯or easier analysis [ or to reduce the cells ] in a chi-square. � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 SNAPSHOT.08 EZESTAT2 旼Page 39컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 쿎hoice of Test � � � 쿟he choice of test depends upon whether the data is measured at � 퀃he CONTINUOUS or the CATEGORICAL (or ORDINAL) level.. � � � � Variable Type of Statistics � � Statistic � �------------------------------------------------------------------------ � � CONTINUOUS PARAMETRIC Means; Standard Deviations � � (e.g. Incomes) Correlation (Pearson) � � Regression � � 't' tests on means, proportions � � � � CATEGORICAL NON- Frequency Distributions � � (e.g. coding PARAMETRIC Correlation (Spearman) � � number for Chi-square � � SEX) Kolmogorov-Smirnov tests � �------------------------------------------------------------------------- � � � � � � � � [Pg Dn] [Pg Up] � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸