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Statistical tests/prosthodontic courses

This document provides an overview of statistical tests and their relevance to dental research. It discusses descriptive statistics such as measures of central tendency (mean, median, mode) and dispersion (standard deviation, variance). It also covers the normal distribution and introduces various parametric and non-parametric tests that can be used to determine statistical significance in dental studies, including t-tests, ANOVA, chi-square tests, and rank-based tests. The goal of statistical testing is to evaluate hypotheses about population parameters based on sample data and reduce the likelihood of making incorrect conclusions.

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Statistical tests and their relevance to Dental Research INDIAN DENTAL ACADEMY Leader in continuing Dental Education www.indiandentalacademy.com
www.indiande ntalacademy.c om Contents Introduction Descriptive statistics Tests of significance Parametric test Non Parametric test Application of tests in dental research Conclusion Bibliography
www.indiande ntalacademy.c om Introduction Normal BP – 120/80 mm Hg Europeans are taller than Asians Average male adult weighs 70kgs Drug A is better than drug B - Endless Cannot be arrived by just Raw data Numbers tell tales – Speak the language of STATISTICS – Adds meaning to data – helps to interpret data Thus lending “significance” to the study
www.indiande ntalacademy.c om Descriptive statistics STATISTICS – Is the science of compiling, classifying & tabulating numerical data and expressing the results in a mathematical or graphical form. OR Statistics is the study of methods & procedures for collecting, classifying, summarizing & analyzing data & for making scientific inferences from such data. - Prof P.V.Sukhatme
www.indiande ntalacademy.c om BIOSTATISTICS –Is the branch of statistics applied to biological or medical sciences (biometry). OR - Is that branch of statistics concerned with mathematical facts and data relating to biological events.
www.indiande ntalacademy.c om VARIABLE – A general term for any feature of the unit which is observed or measured. FREQUENCY DISTRIBUTION – Distribution showing the number of observations or frequencies at different values or within certain range of values of the variable.
www.indiande ntalacademy.c om Mean Dividing the total of all observations by the number of observations n x n xxxx x n ∑ = ++++ = ....321 4.2 5 12 5 0.23.02.72.02.3 0,2.0.2.0,2.7,3.2.3,scoresDMFTofmeanthecalculateEg. == ++++ =x
www.indiande ntalacademy.c om Geometric mean (GM) – nth root of the product ii x n xn HM 111 1 ∑ = ∑ = ( )( )( ) ( ) n xn xxxxGM n log ...321 ∑ == When the variation between the lowest and the highest value is very high, geometric mean is advised & preferred Harmonic mean (HM) – is the reciprocal of the arithmetic mean of the reciprocal of the observations
www.indiande ntalacademy.c om Median – is the middle value, which divides the observed values into two equal parts, when the values are arranged in ascending or descending order     + 2 1n 2.33 2 615 3,2.7,3.02.0,2.0.2. order,ascin 0,2.0.2.0,2.7,3.2.3,scoresDMFTofmedianthecalculateEg. ie arrange valuerd == =+ =
www.indiande ntalacademy.c om Mode – is the value of the variable which occurs most frequently Mode = (3median – 2mean) 1.24.222.33Mode 2.0Mode 0,2.0.2.0,2.7,3.2.3,scoresDMFTofmodethecalculateEg. =×−×= =
www.indiande ntalacademy.c om Variance Is the appropriate measure of dispersion for interval or ratio level data Computes how far each score is from the mean Each score will have a deviation from the mean, so to find the average deviation => we have to add all the deviations and divide it by number of scores (just like calculating mean) This is done by ( )xx −
www.indiande ntalacademy.c om ( ) ( ) 0.... .. =− − ∑ ∑ xxbut N xx ei So to eliminate this zero, square the deviations which eliminates the (-) sign ( ) 2 2 .. S N xx ei = −∑ - is the average of the squared deviations
www.indiande ntalacademy.c om Standard deviation(Root Mean Square deviation)  Is defined as the square root of the arithmetic mean of the squared deviations of the individual values from their arithmetic mean ( ) ( ) N xx sSD 2 −∑ == ( ) ( ) 1 2 − −∑ == N xx SD σ For small samples For large samples
www.indiande ntalacademy.c om ( ) ( ) 1 2 − −∑ == N xxf SD σ ( ) ( ) N xxf sSD 2 −∑ == For small samples For large samples For frequency distribution
www.indiande ntalacademy.c om Uses of SD 1. Summarizes the deviations of a large distribution from mean in one figure used as unit of freedom 2. Indicates whether the variation from the mean is by chance or real 3. Helps finding standard error- which determines whether the difference b/n means of two samples is by chance or real 4. Helps finding the suitable size of the sample for valid conclusions
www.indiande ntalacademy.c om Standard error Standard deviation of mean values Used to compare means with one another n SD SE == sizesample deviationStandard
www.indiande ntalacademy.c om Coefficient of variation is a measure used to measure relative variability I.e, Variation of same character in two or more different series . (eg – pulse rate in young & old person) Variation of two different characters in one & same series . (eg – height & weight in same individual) 100 Mean DeviationStandard ×=CV
www.indiande ntalacademy.c om Normal curve and distribution The histogram of the same frequency distribution of heights, with large number of observations & small class intervals – gives a frequency curve which is symmetrical in nature  Normal curve or Gaussian curve
www.indiande ntalacademy.c om x Normal curve
www.indiande ntalacademy.c om Characteristics of normal curve Bell shaped Symmetrical Mean, Mode & Median – coincide Has two inflections – the central part is convex, while at the point of inflection the curve changes from convexity to concavity
www.indiande ntalacademy.c om  On preparing frequency distribution with small class intervals of the data collected, we can observe A distribution of this nature or shape is called Normal or Gaussian distribution 1. Some observations are above the mean & others are below the mean 2. If arranged in order, maximum number of frequencies are seen in the middle around the mean & fewer at the extremes decreasing smoothly 3. Normally half the observations lie above & half below the mean & all are symmetrically distributed on each side of mean
www.indiande ntalacademy.c om Arithmetically nsobservatio68.27%include,limits1SDmean ± nsobservatio95.45%includelimits,2SDmean ± nsobservatio95%includelimits,96.1 SDmean ± nsobservatio99.73%includeslimits,3SDmean ± nsobservatio99%includeslimits,58.2 SDmean ±
www.indiande ntalacademy.c om x 68.26% 95.46% 99.72% SDx 1− SDx 1+ SDx 2− SDx 2+ SDx 3− SDx 3+ Normal curve and distribution
142.5 3 145.0 8 147.5 15 150.0 45 152.5 90 155.0 155 157.5 194 160.0(M) 195 162.5 136 165.0 93 167.5 42 170.0 16 172.5 6 175.0-177.5 2 Mean – 160.0 SD – 5cm Height in cm frequency of each group frequency with in height limits of Mean ±1SD 680 68% Mean ±2SD 950 95% Mean ±3SD 995 99% www.indiande ntalacademy.c om
www.indiande ntalacademy.c om Relative or standard normal deviate or variate – (Z) Is the deviation from the mean in a normal distribution or curve -in standard normal curve the mean is taken as zero & SD as unity of one SD xx SD Z − == mean-nobservatio
www.indiande ntalacademy.c om Skewness Skewness – as the static to measure the asymmetry coefficient of skewness is 0 Bimodal Negatively (left) skewed Positively (right) skewed
www.indiande ntalacademy.c om kurtosis Kurtosis – is a measure of height of the distribution curve Coefficient of kurtosis is 3 Mesokurtic (normal) Platykurtic (flat) Leptokurtic(high)
www.indiande ntalacademy.c om Tests of significance Population is any finite collection of elements I.e – individuals, items, observations etc,. Statistic – is a quantity describing a sample, namely a function of observations Parameter –  is a constant describing a population Sample – is a part or subset of the population
www.indiande ntalacademy.c om Statistic (Greek) Parameter (Latin) Mean Standard Deviation Variance Correlation coefficient Number of subjects x s 2 s r n µ σ 2 σ ρ N
www.indiande ntalacademy.c om Hypothesis testing Hypothesis is an assumption about the status of a phenomenon or is a statement about the parameters or form of population 0H H Null hypothesis or hypothesis of no difference – States no difference between statistic of a sample & parameter of population or b/n statistics of two samples This nullifies the claim that the experiment result is different from or better than the one observed already Denoted by
www.indiande ntalacademy.c om Alternate hypothesis – Any hypothesis alternate to null hypothesis, which is to be tested Denoted by 1H Note : the alternate hypothesis is accepted when null hypothesis is rejected
www.indiande ntalacademy.c om Type I & type II errors Type I error = Type II error = No errorType II erroris true Type I errorNo erroris true AcceptAccept 0H 1H 1H 0H α β When primary concern of the test is to see whether the null hypothesis can be rejected such test is called Test of significance
www.indiande ntalacademy.c om The probability of committing type I error is called “P” value Thus p-value is the chance that the presence of difference is concluded when actually there is none α Type I error – important- fixed in advance at a low level – such upper limit of tolerance of the chance of type I error is called Level of Significance ( ) Thus α is the maximum tolerable probability of type I error
www.indiande ntalacademy.c om Difference b/n level of significance & P- value - LOS P-value 1) Maximum tolerable chance of type I error 1) Actual probability of type I error 2) α is fixed in advance 2) calculated on basis of data following procedures The P-value can be more than α or less than α depending on data When P-value is < than α  results is statistically significant
www.indiande ntalacademy.c om The level of significance is usually fixed at 5% (0.05) or 1% (0.01) or 0.1% (0.001) or 0.5% (0.005) Maximum desirable is 5% level When P-value is b/n 0.05-0.01 = statistically significant < than 0.01= highly statistically significant Lower than 0.001 or 0.005 = very highly significant
www.indiande ntalacademy.c om 1/2α 1/2 α Zone of Rejection H0 Zone of Rejection H0 Zone of Acceptance H0 SDx 96.1+SDx 96.1− x Sampling Distribution Confidence limits – 95% Confidence interval
www.indiande ntalacademy.c om Tests of significance  Are mathematical methods by which the probability (P) or relative frequency of an observed difference, occurring by chance is found  Steps & procedure of test of significance – 1. State null hypothesis 2. State alternate hypothesis 3. Selection of the appropriate test to be utilized & calculation of test criterion based on type of test 0H 1H
www.indiande ntalacademy.c om 4. Fixation of level of significance 5. Select the table & compare the calculated value with the critical value of the table 6. If calculated value is > table value, is rejected 7. If calculated value is < table value, is accepted 8. Draw conclusions 0H 0H
TESTS IN TEST OF SIGNIFICANCE Parametric (normal distribution & Normal curve ) Non-parametric (not follow normal distribution) Quantitative data Qualitative data Student ‘t’ test 1) Paired 2) Unpaired Z test (for large samples) One way ANOVA Two way ANOVA 1) Z – prop testQualitative (quantitative converted to qualitative ) 1. Mann Whitney U test 2. Wilcoxon rank test 3. Kruskal wallis test 4. Spearman test www.indiande ntalacademy.c om
www.indiande ntalacademy.c om Parametric Uses Non-parametric Paired t test Test of diff b/n Paired observation Wilcoxon signed rank test Two sample t test Comparison of two groups Wilcoxon rank sum test Mann Whitney U test Kendall’s s test One way Anova Comparison of several groups Kruskal wallis test Two way Anova Comparison of groups values on two variables Friedmann test Correlation coefficient Measure of association B/n two variable Spearman’s rank Correlation Kendall’s rank correlation Normal test (Z test ) Chi square test
www.indiande ntalacademy.c om Student ‘t’ test Small samples do not follow normal distribution as the large ones do => will not give correct results Prof W.S.Gossett – Student‘t’ test – pen name – student It is the ratio of observed difference b/n two mean of small samples to the SE of difference in the same
www.indiande ntalacademy.c om Actually, t-value  Z-value of large samples, but the probability (P) of this is determined by reference ‘t’ table Degree of freedom (df)- is the quantity in the denominator which is one less than independent number of observations in a sample For unpaired ‘t’ test = For paired ‘t’ test = n-1 221 −+ nn Types Unpaired ‘t’ test Paired ‘t’ test
www.indiande ntalacademy.c om  Criteria for applying ‘t’ test –  Random samples  Quantitative data  Variable follow normal distribution  Sample size less than 30  Application of ‘t’ test – 1. Two means of small independent sample 2. Sample mean and population mean 3. Two proportions of small independent samples
www.indiande ntalacademy.c om Unpaired ‘t’ test I) Difference b/n means of two independent samples Group 1 Group 2 Sample size Mean SD 1n 2n 1x 2x 1SD 2SD ( ) 0210 =−⇒ xxH ( ) 0211 ≠−⇒ xxH 1) Null hypothesis 2) Alternate hypothesis Data –
www.indiande ntalacademy.c om 3) Test criterion ( )21 21 xxSE xx t − − = ( ) bycalculatedisofhere 21 xxSE − ( )       +=− 21 21 11 of nn SDxxSE ( ) ( ) 2 11 where 21 2 22 2 11 −+ −+− = nn SDnSDn SD ( ) ( ) ( )       + −+ −+− =− 2121 2 22 2 11 21 11 2 11 nnnn SDnSDn xxSE
www.indiande ntalacademy.c om 4) Calculate degree of freedom ( ) ( ) 111 2121 −+=−+−= nnnndf 6) Draw conclusions 5) Compare the calculated value & the table value
www.indiande ntalacademy.c om Example – difference b/n caries experience of high & low socioeconomic group Sl no Details High socio economic group Low socio economic group I Sample size II DMFT III Standard deviation 151 =n 102 =n 91.21 =x 26.22 =x 27.01 =SD 22.02 =SD ( ) 23,34.6 1027.0 65.0 21 21 === − − = df xxSE xx t 001.0001.0 76.3 ttt c >∴= There is a significant difference
www.indiande ntalacademy.c om T table
www.indiande ntalacademy.c om Other applications II) Difference b/n sample mean & population mean n SDSE x t = − = µ 1−= ndf       + − = 21 21 11 nn PQ pp t 21 2211 where nn pnpn P + + = PQ −=1 221 −+= nndf III) Difference b/n two sample proportions
www.indiande ntalacademy.c om Paired ‘t’ test  Is applied to paired data of observations from one sample only when each individual gives a paired of observations  Here the pair of observations are correlated and not independent, so for application of ‘t’ test following procedure is used- 1. Find the difference for each pair 2. Calculate the mean of the difference (x) ie 3. Calculate the SD of the differences & later SE xyy =− 21 x       = n SD SE
www.indiande ntalacademy.c om 4. Test criterion ( ) ( ) n xSD x dSE x t = − = 0 1−= ndf 7. Draw conclusions 6. Refer ‘t’ table & find the probability of calculated value 5. Degree of freedom
www.indiande ntalacademy.c om Example – to find out if there is any significant improvement in DAI scores before and after orthodontic treatment Sl no DAI before DAI after Difference Squares 1 30 24 6 36 2 26 23 3 9 3 27 24 3 9 4 35 25 10 100 5 25 23 2 4 Total 20 158
www.indiande ntalacademy.c om ( ) ( ) ( ) ( ) ( ) ( )222222 42410434346squares,ofsum 4 5 20 −+−+−+−+−=− === ∑ ∑ xx n x xMean 46 436114 = ++++= ( ) 78.2but 416352.2 5179.1 4 5179.1 5 391.3 391.35.11 4 46 1 5.0 5.0 2 tt t ndf SE x t n SD SE n xx SD c c <∴ = =−====∴ ===∴ === − − = ∑ Hence not significant
www.indiande ntalacademy.c om Z test (Normal test) Similar to ‘t’ test in all aspect except that the sample size should be > 30 In case of normal distribution, the tabulated value of Z at - 960.1level%5 05.0 == Z 576.2level%1 01.0 == Z 290.3level%1.0 001.0 == Z
www.indiande ntalacademy.c om  Z test can be used for – 1. Comparison of means of two samples –       += 2 2 2 1 2 1 n SD n SD ( )21 21 xxSE xx Z − − = ( ) ( )2 2 2 121where SESExxSE +=− n SD x Z 2 µ− = 2. Comparison of sample mean & population mean
www.indiande ntalacademy.c om 3. Difference b/n two sample proportions 21 2211 21 21 herew 11 nn pnpn P nn PQ pp Z + + =             + − = PQ −=1             − = n PQ Pp Z 1 Where p = sample proportion P = populn proportion 4. Comparison of sample proportion (or percentage) with population proportion (or percentage)
www.indiande ntalacademy.c om Analysis of variance (ANOVA)  Useful for comparison of means of several groups  Is an extension of student’s ‘t’ test for more than two groups  R A Fisher in 1920’s  Has four models 1. One way classification (one way ANOVA ) 2. Single factor repeated measures design 3. Nested or hierarchical design 4. Two way classification (two way ANOVA)
www.indiande ntalacademy.c om One way ANOVA Can be used to compare like- Effect of different treatment modalities Effect of different obturation techniques on the apical seal , etc,.
www.indiande ntalacademy.c om Groups (or treatments) 1 2 i k Individual values Calculate No of observations Sum of x values Sum of squares Mean of values 11x 2ix22x nx2nx1 12x 1kx1ix21x inx 2kx knx n n n n nxxx 11211 ...+++=1Τ 2T iT kT ( ) ( ) ( )2 1 2 12 2 11 .. nxxx +++=1S 2S iS kS n T x 1 1 = 2x ix kx
www.indiande ntalacademy.c om ANOVA table Sl no Source of variation Degree of freedom Sum of squares Mean sum of squares F ratio or variance ratio I Between Groups II With in groups III Total 1−k kn − 1−n ( )       −=− ∑∑ i ii i N T xxx 2 22 ( ) ∑∑ ∑∑ ∑ −=− i i i i j iji j iij n T xxx 2 22 ( )       −=− ∑ ∑∑ ∑ N T xxx i j iji j ij 2 22 ( ) 1 2 2 − − = ∑ k xx S i i B kN n T x S i j i i i ij W −       − = ∑ ∑ ∑ 2 2 2 1 22 2 − − = ∑ ∑ N N Tx S i j ij T ( )kNk S S W B −− ,12 2
ANOVA www.indiande ntalacademy.c om
www.indiande ntalacademy.c om Example- see whether there is a difference in number of patients seen in a given period by practitioners in three group practice Practice A B C Individual values 268 387 161 349 264 346 328 423 324 209 254 293 292 239 Calculate No of observations (n) 5 4 5 Sum of x values 1441 1328 1363 Sum of squares 426899 462910 393583 Mean of values 288.2 332.0 272.6
www.indiande ntalacademy.c om ( ) 71.63861 2 222 = ++ ++ −++= ∑∑ CBA CBA CBA nnn xxx xxx ( ) ( ) ( ) ( ) 71.8215 2222 = ++ ++ −++= ∑ ∑∑∑∑∑ CBA CBA C C B B A A nnn xxx n x n x n x 55646.0 SSbetween-SStotal = = Between group sum of squares Total sum of squares With in group sum of squares
www.indiande ntalacademy.c om ANOVA table Sl no Source of variation Degree of freedom Sum of squares Mean sum of squares F ratio or variance ratio I Between Groups II With in groups III Total 213 =− 11314 =− 13114 =− 71.8215 71.63861 55646.0 86.4107 2 71.8215 = 73.5088 11 55646 = 81.0 73.5088 86.4107 = ( )11,298.381.0 05.0 === dfFF Because FC < FT , there is no significant difference in the number of patients attending 3 different practice
www.indiande ntalacademy.c om Further, any particular pair of treatments can be compared using SE of difference b/n two means Eg – cd xx & ( ) criteriontestt'' usingbytestedbemaydifference& 11 cd cd cd xx nn MSExxSE −       +=− ( )cd cd xxSE xx t − − =
www.indiande ntalacademy.c om Two way ANOVA Is used to study the impact of two factors on variations in a specific variable Eg – Effect of age and sex on DMFT value
Sample values blocks Treatments sample size Total Mean value i ii .. n Sample size Total Mean value 11x 32x22x nx2nx1 12x 1kx31x21x nx3 2kx knx n n n n k k k Nnk = 2T nT 1T 1T 2T kT3T T 2x 1x kx3x2x1x nx xwww.indiande ntalacademy.c om
www.indiande ntalacademy.c om Sl no Source Sum of squares Degree of freedom Mean sum of squares (MSS) Variance ratio F I Blocks II Treatments III Residual or error IV Total 2 way ANOVA table 1−n 1−k ( )( )11 −− kn ( ) 11 −=− Nnk 1 blocks − = n SS MS blocks 1− = k SS MS treatments treatments ( )( )11 −− = kn SS MS residual residual residual blocks MS MS F =1 residual treatment MS MS F =2 ( ) ( )( )11Vs1ofwithblocksofratiovariance1 −−−= knndfF ( ) ( )( )11Vs1ofwiths'treatmentofratiovariance2 −−−= knkdfF blocksSS treatmentsSS residualSS totalSS
www.indiande ntalacademy.c om Multiple comparison tests 1. Fisher’s procedure – student’s ‘t’ test 2. Least significant difference method (LSD)  Just like student’s ‘t’ test  To test significant difference b/n two groups or variable means 1. Scheffe’s significant difference procedure  Is applicable when groups having heterogeneous variance or variations 1. Tukey’s method  For comparison of the differences b/n all possible pairs of treatments or group means
www.indiande ntalacademy.c om 5. Duncan’s multiple comparison test – For all comparisons of paired groups only 6. Dunnet’s comparison test procedure – For comparison of one control and several treatment groups
www.indiande ntalacademy.c om Non parametric tests Here the distribution do not require any specific pattern of distribution. They are applicable to almost all kinds of distribution Chi square test Mann Whitney U test Wilcoxon signed rank test Wilcoxon rank sum test Kendall’s S test Kruskal wallis test Spearman’s rank correlation
www.indiande ntalacademy.c om Chi square test  By Karl Pearson & denoted as χ²  Application 1. Alternate test to find the significance of difference in two or more than two proportions 2. As a test of association b/n two events in binomial or multinomial samples 3. As a test of goodness of fit
www.indiande ntalacademy.c om Requirement to apply chi square test Random samples Qualitative data Lowest observed frequency not less than 5 Contingency table Frequency table where sample classified according to two different attributes 2 rows ; 2 columns => 2 X 2 contingency table r rows : c columns => rXc contingency table ( ) ∑ − = E EO 2 2 χ O – observed frequency E – expected frequency
www.indiande ntalacademy.c om  Steps 1. State null & alternate hypothesis 2. Make contingency table of the data 3. Determine expected frequency by 4. Calculate chi-square of each by- ( )cr × ( )frequencytotalN cr E × = ( ) E EO 2 2 − =χ
www.indiande ntalacademy.c om 5. calculate degree of freedom 6. Sum all the chi-square of each cell – this gives chi-square value of the data 7. Compare the calculated value with the table value at any LOS 8. Draw conclusions ( ) ∑ − = E EO 2 2 χ ( )( )11 −−= rcdf
www.indiande ntalacademy.c om Example from a dental health campaign School Oral hygiene Total G F+ F- P Below avg 62 (85.9) 103 (93.0) 57 (45.2) 11 (8.9) 233 Avg 50 (43.9) 36 (47.5) 26 (23.1) 7 (4.6) 119 Above avg 80 (62.3) 69 (67.5) 18 (32.8) 2 (6.5) 169 Total 192 208 101 20 521 ( )frequencytotalN cr E × = ( )( ) 63211 =×=−−= rcdf ( ) 4.31 2 2 = − = ∑ E EO cχ 22.46is0.001PatTable 2 =tχ Hence significant difference
www.indiande ntalacademy.c om
www.indiande ntalacademy.c om Alternate formulae If we have contingency table ( ) ( )( )( )( ) 1with 2 2 = ++++ − = df dbcadcba bcadN χ ( )( )( )( ) 1with 2 2 2 = ++++       −− = df dbcadcba N bcadN χ a b a+b c d c+d a+c b+d a+b+c+d=N If one of the value is below 5 => Yate’s correction formula
www.indiande ntalacademy.c om If the table is larger than 2X2, Yate’s correction cannot be applied – then the small frequency (<5) can be pooled or combined with next group or class in the table Chi square test only tells the presence or absence of association, but does not measure the strength of association
www.indiande ntalacademy.c om  If degree of association as to be calculated then – 1. Yule’s coefficient of association bcad bcad Q + − = ad bc ad bc Y + − = 1 1 ( )( )( )( )dcdbcaba bcad V ++++ − = 2 2 χ χ + = N C 4. Pearson’s coefficient of contingency 3. - 2. Yule’s coefficient of colligation
www.indiande ntalacademy.c om Wilcoxon signed rank test Is equivalent to paired ‘t’ test Steps Exclude any differences which are zero Put the remaining differences in ascending order, ignoring the signs Gives ranks from lowest to highest If any differences are equal, then average their ranks Count all the ranks of positive differences – T+ Count all the ranks of negative differences – T-
www.indiande ntalacademy.c om If there is no differences b/n variables then T+ & T_ will be similar, but if there is difference then one sum will be large and the other will be much smaller T= smaller of T+&T_ Compare the T value with the critical value for 5%, 2% & 1% significance level A result is significant if it is smaller than critical value
www.indiande ntalacademy.c om Example:Results of a placebo-controlled clinical trail to test the effectiveness of sleeping drug Patients Sleep hrs Drug Placebo 1 6.1 5.2 2 7.0 7.9 3 8.2 3.9 4 7.6 4.7 5 6.5 5.3 6 8.4 5.4 7 6.9 4.2 8 6.7 6.1 9 7.4 3.8 10 5.8 6.3 Difference 0.9 -0.9 4.3 2.9 1.2 3.0 2.7 0.6 3.6 -0.5 Rank with signs + - 3.5 - - -3.5 10 - 7 - 5 - 8 - 6 - 2 - 9 - - -1 50.5 -4.5
www.indiande ntalacademy.c om Calculated T=-4.5 df =10, Table value at 5% (n=10)= 8 Cal T< table value, H0 is rejected We conclude that sleeping drug is more effective than the placebo
www.indiande ntalacademy.c om Mann Whitney U test Is used to determine whether two independent sample have been drawn from same sample It is a alternative to student ‘t’ test & requires at least ordinal or normal measurement ( ) 21 11 21 2 1 RorR nn nnU − + += Where, n1n2 are sample sizes R1 R2 are sum of ranks assigned to I & II group
www.indiande ntalacademy.c om All the observation in two samples are ranked numerically from smallest to largest without regarding the groups Procedure Then identify the observation for I and II samples Sum of ranks for I and II sample determined separately Take difference of two sum T =R1 - R2
www.indiande ntalacademy.c om Comparison of birth weights of children born to 15 non smokers with those of children born to 14 heavy smokers NS 3.9 3.7 3.6 3.7 3.2 4.2 4.0 3.6 3.8 3.3 4.1 3.2 3.5 3.5 2.7 HS 3.1 2.8 2.9 3.2 3.8 3.5 3.2 2.7 3.6 3.7 3.6 2.3 2.3 3.6 R1 26 23 16 21 8 29 27 17 24 12 28 10 15 13 03 R2 7 5 6 11 25 14 9 4 20 22 19 2 1 18 Ranks assignments
www.indiande ntalacademy.c om Sum of R1= 272 and Sum of R2=163 Difference T=R1 – R2 is 109 The table value of T0.05 is 96 , so reject the H0 We conclude that weights of children born to the heavy smokers are significantly lower than those of the children born to the non-smokers (p<0.05)
www.indiande ntalacademy.c om Applications of statistical tests in Research Methods
www.indiande ntalacademy.c om One variable, two group sample problem
www.indiande ntalacademy.c om One variable, multiple group sample problem
www.indiande ntalacademy.c om Interested in relationship b/n two variables Are both continuous variablesUse linear regression Is one variable continuous & other categorical Use analysis of variance Or Logistic regression Both variables categorical Use contingency table analysis Two variable problem Yes No Yes No
www.indiande ntalacademy.c om Research interested in relationship B/n more than two variables Use multiple regression Or Multivariate analysis Multiple – variable problem
www.indiande ntalacademy.c om Conclusion Statistics are excellent tools in research data analysis; how ever, if inappropriately used they may make the results of a well conducted research study un-interpretable or meaningless
www.indiande ntalacademy.c om
www.indiande ntalacademy.c om

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Statistical tests/prosthodontic courses

  • 1.
    Statistical tests andtheir relevance to Dental Research INDIAN DENTAL ACADEMY Leader in continuing Dental Education www.indiandentalacademy.com
  • 2.
    www.indiande ntalacademy.c om Contents Introduction Descriptive statistics Tests ofsignificance Parametric test Non Parametric test Application of tests in dental research Conclusion Bibliography
  • 3.
    www.indiande ntalacademy.c om Introduction Normal BP –120/80 mm Hg Europeans are taller than Asians Average male adult weighs 70kgs Drug A is better than drug B - Endless Cannot be arrived by just Raw data Numbers tell tales – Speak the language of STATISTICS – Adds meaning to data – helps to interpret data Thus lending “significance” to the study
  • 4.
    www.indiande ntalacademy.c om Descriptive statistics STATISTICS – Isthe science of compiling, classifying & tabulating numerical data and expressing the results in a mathematical or graphical form. OR Statistics is the study of methods & procedures for collecting, classifying, summarizing & analyzing data & for making scientific inferences from such data. - Prof P.V.Sukhatme
  • 5.
    www.indiande ntalacademy.c om BIOSTATISTICS –Is the branchof statistics applied to biological or medical sciences (biometry). OR - Is that branch of statistics concerned with mathematical facts and data relating to biological events.
  • 6.
    www.indiande ntalacademy.c om VARIABLE – A generalterm for any feature of the unit which is observed or measured. FREQUENCY DISTRIBUTION – Distribution showing the number of observations or frequencies at different values or within certain range of values of the variable.
  • 7.
    www.indiande ntalacademy.c om Mean Dividing the totalof all observations by the number of observations n x n xxxx x n ∑ = ++++ = ....321 4.2 5 12 5 0.23.02.72.02.3 0,2.0.2.0,2.7,3.2.3,scoresDMFTofmeanthecalculateEg. == ++++ =x
  • 8.
    www.indiande ntalacademy.c om Geometric mean (GM)– nth root of the product ii x n xn HM 111 1 ∑ = ∑ = ( )( )( ) ( ) n xn xxxxGM n log ...321 ∑ == When the variation between the lowest and the highest value is very high, geometric mean is advised & preferred Harmonic mean (HM) – is the reciprocal of the arithmetic mean of the reciprocal of the observations
  • 9.
    www.indiande ntalacademy.c om Median – is themiddle value, which divides the observed values into two equal parts, when the values are arranged in ascending or descending order     + 2 1n 2.33 2 615 3,2.7,3.02.0,2.0.2. order,ascin 0,2.0.2.0,2.7,3.2.3,scoresDMFTofmedianthecalculateEg. ie arrange valuerd == =+ =
  • 10.
    www.indiande ntalacademy.c om Mode – is thevalue of the variable which occurs most frequently Mode = (3median – 2mean) 1.24.222.33Mode 2.0Mode 0,2.0.2.0,2.7,3.2.3,scoresDMFTofmodethecalculateEg. =×−×= =
  • 11.
    www.indiande ntalacademy.c om Variance Is the appropriatemeasure of dispersion for interval or ratio level data Computes how far each score is from the mean Each score will have a deviation from the mean, so to find the average deviation => we have to add all the deviations and divide it by number of scores (just like calculating mean) This is done by ( )xx −
  • 12.
    www.indiande ntalacademy.c om ( ) ( )0.... .. =− − ∑ ∑ xxbut N xx ei So to eliminate this zero, square the deviations which eliminates the (-) sign ( ) 2 2 .. S N xx ei = −∑ - is the average of the squared deviations
  • 13.
    www.indiande ntalacademy.c om Standard deviation(Root Mean Squaredeviation)  Is defined as the square root of the arithmetic mean of the squared deviations of the individual values from their arithmetic mean ( ) ( ) N xx sSD 2 −∑ == ( ) ( ) 1 2 − −∑ == N xx SD σ For small samples For large samples
  • 14.
    www.indiande ntalacademy.c om ( ) () 1 2 − −∑ == N xxf SD σ ( ) ( ) N xxf sSD 2 −∑ == For small samples For large samples For frequency distribution
  • 15.
    www.indiande ntalacademy.c om Uses of SD 1.Summarizes the deviations of a large distribution from mean in one figure used as unit of freedom 2. Indicates whether the variation from the mean is by chance or real 3. Helps finding standard error- which determines whether the difference b/n means of two samples is by chance or real 4. Helps finding the suitable size of the sample for valid conclusions
  • 16.
    www.indiande ntalacademy.c om Standard error Standard deviationof mean values Used to compare means with one another n SD SE == sizesample deviationStandard
  • 17.
    www.indiande ntalacademy.c om Coefficient of variation isa measure used to measure relative variability I.e, Variation of same character in two or more different series . (eg – pulse rate in young & old person) Variation of two different characters in one & same series . (eg – height & weight in same individual) 100 Mean DeviationStandard ×=CV
  • 18.
    www.indiande ntalacademy.c om Normal curve anddistribution The histogram of the same frequency distribution of heights, with large number of observations & small class intervals – gives a frequency curve which is symmetrical in nature  Normal curve or Gaussian curve
  • 19.
  • 20.
    www.indiande ntalacademy.c om Characteristics of normalcurve Bell shaped Symmetrical Mean, Mode & Median – coincide Has two inflections – the central part is convex, while at the point of inflection the curve changes from convexity to concavity
  • 21.
    www.indiande ntalacademy.c om  On preparingfrequency distribution with small class intervals of the data collected, we can observe A distribution of this nature or shape is called Normal or Gaussian distribution 1. Some observations are above the mean & others are below the mean 2. If arranged in order, maximum number of frequencies are seen in the middle around the mean & fewer at the extremes decreasing smoothly 3. Normally half the observations lie above & half below the mean & all are symmetrically distributed on each side of mean
  • 22.
  • 23.
    www.indiande ntalacademy.c om x 68.26% 95.46% 99.72% SDx 1− SDx1+ SDx 2− SDx 2+ SDx 3− SDx 3+ Normal curve and distribution
  • 24.
    142.5 3 145.0 8 147.515 150.0 45 152.5 90 155.0 155 157.5 194 160.0(M) 195 162.5 136 165.0 93 167.5 42 170.0 16 172.5 6 175.0-177.5 2 Mean – 160.0 SD – 5cm Height in cm frequency of each group frequency with in height limits of Mean ±1SD 680 68% Mean ±2SD 950 95% Mean ±3SD 995 99% www.indiande ntalacademy.c om
  • 25.
    www.indiande ntalacademy.c om Relative or standardnormal deviate or variate – (Z) Is the deviation from the mean in a normal distribution or curve -in standard normal curve the mean is taken as zero & SD as unity of one SD xx SD Z − == mean-nobservatio
  • 26.
    www.indiande ntalacademy.c om Skewness Skewness – asthe static to measure the asymmetry coefficient of skewness is 0 Bimodal Negatively (left) skewed Positively (right) skewed
  • 27.
    www.indiande ntalacademy.c om kurtosis Kurtosis – isa measure of height of the distribution curve Coefficient of kurtosis is 3 Mesokurtic (normal) Platykurtic (flat) Leptokurtic(high)
  • 28.
    www.indiande ntalacademy.c om Tests of significance Population isany finite collection of elements I.e – individuals, items, observations etc,. Statistic – is a quantity describing a sample, namely a function of observations Parameter –  is a constant describing a population Sample – is a part or subset of the population
  • 29.
  • 30.
    www.indiande ntalacademy.c om Hypothesis testing Hypothesis is anassumption about the status of a phenomenon or is a statement about the parameters or form of population 0H H Null hypothesis or hypothesis of no difference – States no difference between statistic of a sample & parameter of population or b/n statistics of two samples This nullifies the claim that the experiment result is different from or better than the one observed already Denoted by
  • 31.
    www.indiande ntalacademy.c om Alternate hypothesis – Anyhypothesis alternate to null hypothesis, which is to be tested Denoted by 1H Note : the alternate hypothesis is accepted when null hypothesis is rejected
  • 32.
    www.indiande ntalacademy.c om Type I &type II errors Type I error = Type II error = No errorType II erroris true Type I errorNo erroris true AcceptAccept 0H 1H 1H 0H α β When primary concern of the test is to see whether the null hypothesis can be rejected such test is called Test of significance
  • 33.
    www.indiande ntalacademy.c om The probability ofcommitting type I error is called “P” value Thus p-value is the chance that the presence of difference is concluded when actually there is none α Type I error – important- fixed in advance at a low level – such upper limit of tolerance of the chance of type I error is called Level of Significance ( ) Thus α is the maximum tolerable probability of type I error
  • 34.
    www.indiande ntalacademy.c om Difference b/n levelof significance & P- value - LOS P-value 1) Maximum tolerable chance of type I error 1) Actual probability of type I error 2) α is fixed in advance 2) calculated on basis of data following procedures The P-value can be more than α or less than α depending on data When P-value is < than α  results is statistically significant
  • 35.
    www.indiande ntalacademy.c om The level ofsignificance is usually fixed at 5% (0.05) or 1% (0.01) or 0.1% (0.001) or 0.5% (0.005) Maximum desirable is 5% level When P-value is b/n 0.05-0.01 = statistically significant < than 0.01= highly statistically significant Lower than 0.001 or 0.005 = very highly significant
  • 36.
    www.indiande ntalacademy.c om 1/2α 1/2 α Zone of RejectionH0 Zone of Rejection H0 Zone of Acceptance H0 SDx 96.1+SDx 96.1− x Sampling Distribution Confidence limits – 95% Confidence interval
  • 37.
    www.indiande ntalacademy.c om Tests of significance Are mathematical methods by which the probability (P) or relative frequency of an observed difference, occurring by chance is found  Steps & procedure of test of significance – 1. State null hypothesis 2. State alternate hypothesis 3. Selection of the appropriate test to be utilized & calculation of test criterion based on type of test 0H 1H
  • 38.
    www.indiande ntalacademy.c om 4. Fixation oflevel of significance 5. Select the table & compare the calculated value with the critical value of the table 6. If calculated value is > table value, is rejected 7. If calculated value is < table value, is accepted 8. Draw conclusions 0H 0H
  • 39.
    TESTS IN TESTOF SIGNIFICANCE Parametric (normal distribution & Normal curve ) Non-parametric (not follow normal distribution) Quantitative data Qualitative data Student ‘t’ test 1) Paired 2) Unpaired Z test (for large samples) One way ANOVA Two way ANOVA 1) Z – prop testQualitative (quantitative converted to qualitative ) 1. Mann Whitney U test 2. Wilcoxon rank test 3. Kruskal wallis test 4. Spearman test www.indiande ntalacademy.c om
  • 40.
    www.indiande ntalacademy.c om Parametric Uses Non-parametric Pairedt test Test of diff b/n Paired observation Wilcoxon signed rank test Two sample t test Comparison of two groups Wilcoxon rank sum test Mann Whitney U test Kendall’s s test One way Anova Comparison of several groups Kruskal wallis test Two way Anova Comparison of groups values on two variables Friedmann test Correlation coefficient Measure of association B/n two variable Spearman’s rank Correlation Kendall’s rank correlation Normal test (Z test ) Chi square test
  • 41.
    www.indiande ntalacademy.c om Student ‘t’ test Smallsamples do not follow normal distribution as the large ones do => will not give correct results Prof W.S.Gossett – Student‘t’ test – pen name – student It is the ratio of observed difference b/n two mean of small samples to the SE of difference in the same
  • 42.
    www.indiande ntalacademy.c om Actually, t-value Z-value of large samples, but the probability (P) of this is determined by reference ‘t’ table Degree of freedom (df)- is the quantity in the denominator which is one less than independent number of observations in a sample For unpaired ‘t’ test = For paired ‘t’ test = n-1 221 −+ nn Types Unpaired ‘t’ test Paired ‘t’ test
  • 43.
    www.indiande ntalacademy.c om  Criteria forapplying ‘t’ test –  Random samples  Quantitative data  Variable follow normal distribution  Sample size less than 30  Application of ‘t’ test – 1. Two means of small independent sample 2. Sample mean and population mean 3. Two proportions of small independent samples
  • 44.
    www.indiande ntalacademy.c om Unpaired ‘t’ test I)Difference b/n means of two independent samples Group 1 Group 2 Sample size Mean SD 1n 2n 1x 2x 1SD 2SD ( ) 0210 =−⇒ xxH ( ) 0211 ≠−⇒ xxH 1) Null hypothesis 2) Alternate hypothesis Data –
  • 45.
    www.indiande ntalacademy.c om 3) Test criterion ()21 21 xxSE xx t − − = ( ) bycalculatedisofhere 21 xxSE − ( )       +=− 21 21 11 of nn SDxxSE ( ) ( ) 2 11 where 21 2 22 2 11 −+ −+− = nn SDnSDn SD ( ) ( ) ( )       + −+ −+− =− 2121 2 22 2 11 21 11 2 11 nnnn SDnSDn xxSE
  • 46.
    www.indiande ntalacademy.c om 4) Calculate degreeof freedom ( ) ( ) 111 2121 −+=−+−= nnnndf 6) Draw conclusions 5) Compare the calculated value & the table value
  • 47.
    www.indiande ntalacademy.c om Example – differenceb/n caries experience of high & low socioeconomic group Sl no Details High socio economic group Low socio economic group I Sample size II DMFT III Standard deviation 151 =n 102 =n 91.21 =x 26.22 =x 27.01 =SD 22.02 =SD ( ) 23,34.6 1027.0 65.0 21 21 === − − = df xxSE xx t 001.0001.0 76.3 ttt c >∴= There is a significant difference
  • 48.
  • 49.
    www.indiande ntalacademy.c om Other applications II) Differenceb/n sample mean & population mean n SDSE x t = − = µ 1−= ndf       + − = 21 21 11 nn PQ pp t 21 2211 where nn pnpn P + + = PQ −=1 221 −+= nndf III) Difference b/n two sample proportions
  • 50.
    www.indiande ntalacademy.c om Paired ‘t’ test Is applied to paired data of observations from one sample only when each individual gives a paired of observations  Here the pair of observations are correlated and not independent, so for application of ‘t’ test following procedure is used- 1. Find the difference for each pair 2. Calculate the mean of the difference (x) ie 3. Calculate the SD of the differences & later SE xyy =− 21 x       = n SD SE
  • 51.
    www.indiande ntalacademy.c om 4. Test criterion () ( ) n xSD x dSE x t = − = 0 1−= ndf 7. Draw conclusions 6. Refer ‘t’ table & find the probability of calculated value 5. Degree of freedom
  • 52.
    www.indiande ntalacademy.c om Example – tofind out if there is any significant improvement in DAI scores before and after orthodontic treatment Sl no DAI before DAI after Difference Squares 1 30 24 6 36 2 26 23 3 9 3 27 24 3 9 4 35 25 10 100 5 25 23 2 4 Total 20 158
  • 53.
    www.indiande ntalacademy.c om ( ) () ( ) ( ) ( ) ( )222222 42410434346squares,ofsum 4 5 20 −+−+−+−+−=− === ∑ ∑ xx n x xMean 46 436114 = ++++= ( ) 78.2but 416352.2 5179.1 4 5179.1 5 391.3 391.35.11 4 46 1 5.0 5.0 2 tt t ndf SE x t n SD SE n xx SD c c <∴ = =−====∴ ===∴ === − − = ∑ Hence not significant
  • 54.
    www.indiande ntalacademy.c om Z test (Normaltest) Similar to ‘t’ test in all aspect except that the sample size should be > 30 In case of normal distribution, the tabulated value of Z at - 960.1level%5 05.0 == Z 576.2level%1 01.0 == Z 290.3level%1.0 001.0 == Z
  • 55.
    www.indiande ntalacademy.c om  Z testcan be used for – 1. Comparison of means of two samples –       += 2 2 2 1 2 1 n SD n SD ( )21 21 xxSE xx Z − − = ( ) ( )2 2 2 121where SESExxSE +=− n SD x Z 2 µ− = 2. Comparison of sample mean & population mean
  • 56.
    www.indiande ntalacademy.c om 3. Difference b/ntwo sample proportions 21 2211 21 21 herew 11 nn pnpn P nn PQ pp Z + + =             + − = PQ −=1             − = n PQ Pp Z 1 Where p = sample proportion P = populn proportion 4. Comparison of sample proportion (or percentage) with population proportion (or percentage)
  • 57.
    www.indiande ntalacademy.c om Analysis of variance(ANOVA)  Useful for comparison of means of several groups  Is an extension of student’s ‘t’ test for more than two groups  R A Fisher in 1920’s  Has four models 1. One way classification (one way ANOVA ) 2. Single factor repeated measures design 3. Nested or hierarchical design 4. Two way classification (two way ANOVA)
  • 58.
    www.indiande ntalacademy.c om One way ANOVA Canbe used to compare like- Effect of different treatment modalities Effect of different obturation techniques on the apical seal , etc,.
  • 59.
    www.indiande ntalacademy.c om Groups (or treatments)1 2 i k Individual values Calculate No of observations Sum of x values Sum of squares Mean of values 11x 2ix22x nx2nx1 12x 1kx1ix21x inx 2kx knx n n n n nxxx 11211 ...+++=1Τ 2T iT kT ( ) ( ) ( )2 1 2 12 2 11 .. nxxx +++=1S 2S iS kS n T x 1 1 = 2x ix kx
  • 60.
    www.indiande ntalacademy.c om ANOVA table Sl no Source of variation Degree of freedom Sum ofsquares Mean sum of squares F ratio or variance ratio I Between Groups II With in groups III Total 1−k kn − 1−n ( )       −=− ∑∑ i ii i N T xxx 2 22 ( ) ∑∑ ∑∑ ∑ −=− i i i i j iji j iij n T xxx 2 22 ( )       −=− ∑ ∑∑ ∑ N T xxx i j iji j ij 2 22 ( ) 1 2 2 − − = ∑ k xx S i i B kN n T x S i j i i i ij W −       − = ∑ ∑ ∑ 2 2 2 1 22 2 − − = ∑ ∑ N N Tx S i j ij T ( )kNk S S W B −− ,12 2
  • 61.
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    www.indiande ntalacademy.c om Example- see whetherthere is a difference in number of patients seen in a given period by practitioners in three group practice Practice A B C Individual values 268 387 161 349 264 346 328 423 324 209 254 293 292 239 Calculate No of observations (n) 5 4 5 Sum of x values 1441 1328 1363 Sum of squares 426899 462910 393583 Mean of values 288.2 332.0 272.6
  • 63.
    www.indiande ntalacademy.c om ( ) 71.63861 2 222 = ++ ++ −++= ∑∑ CBA CBA CBA nnn xxx xxx ( )( ) ( ) ( ) 71.8215 2222 = ++ ++ −++= ∑ ∑∑∑∑∑ CBA CBA C C B B A A nnn xxx n x n x n x 55646.0 SSbetween-SStotal = = Between group sum of squares Total sum of squares With in group sum of squares
  • 64.
    www.indiande ntalacademy.c om ANOVA table Sl no Source of variation Degreeof freedom Sum of squares Mean sum of squares F ratio or variance ratio I Between Groups II With in groups III Total 213 =− 11314 =− 13114 =− 71.8215 71.63861 55646.0 86.4107 2 71.8215 = 73.5088 11 55646 = 81.0 73.5088 86.4107 = ( )11,298.381.0 05.0 === dfFF Because FC < FT , there is no significant difference in the number of patients attending 3 different practice
  • 65.
    www.indiande ntalacademy.c om Further, any particularpair of treatments can be compared using SE of difference b/n two means Eg – cd xx & ( ) criteriontestt'' usingbytestedbemaydifference& 11 cd cd cd xx nn MSExxSE −       +=− ( )cd cd xxSE xx t − − =
  • 66.
    www.indiande ntalacademy.c om Two way ANOVA Isused to study the impact of two factors on variations in a specific variable Eg – Effect of age and sex on DMFT value
  • 67.
    Sample values blocks Treatmentssample size Total Mean value i ii .. n Sample size Total Mean value 11x 32x22x nx2nx1 12x 1kx31x21x nx3 2kx knx n n n n k k k Nnk = 2T nT 1T 1T 2T kT3T T 2x 1x kx3x2x1x nx xwww.indiande ntalacademy.c om
  • 68.
    www.indiande ntalacademy.c om Sl no Source Sum of squares Degreeof freedom Mean sum of squares (MSS) Variance ratio F I Blocks II Treatments III Residual or error IV Total 2 way ANOVA table 1−n 1−k ( )( )11 −− kn ( ) 11 −=− Nnk 1 blocks − = n SS MS blocks 1− = k SS MS treatments treatments ( )( )11 −− = kn SS MS residual residual residual blocks MS MS F =1 residual treatment MS MS F =2 ( ) ( )( )11Vs1ofwithblocksofratiovariance1 −−−= knndfF ( ) ( )( )11Vs1ofwiths'treatmentofratiovariance2 −−−= knkdfF blocksSS treatmentsSS residualSS totalSS
  • 69.
    www.indiande ntalacademy.c om Multiple comparison tests 1.Fisher’s procedure – student’s ‘t’ test 2. Least significant difference method (LSD)  Just like student’s ‘t’ test  To test significant difference b/n two groups or variable means 1. Scheffe’s significant difference procedure  Is applicable when groups having heterogeneous variance or variations 1. Tukey’s method  For comparison of the differences b/n all possible pairs of treatments or group means
  • 70.
    www.indiande ntalacademy.c om 5. Duncan’s multiplecomparison test – For all comparisons of paired groups only 6. Dunnet’s comparison test procedure – For comparison of one control and several treatment groups
  • 71.
    www.indiande ntalacademy.c om Non parametric tests Herethe distribution do not require any specific pattern of distribution. They are applicable to almost all kinds of distribution Chi square test Mann Whitney U test Wilcoxon signed rank test Wilcoxon rank sum test Kendall’s S test Kruskal wallis test Spearman’s rank correlation
  • 72.
    www.indiande ntalacademy.c om Chi square test By Karl Pearson & denoted as χ²  Application 1. Alternate test to find the significance of difference in two or more than two proportions 2. As a test of association b/n two events in binomial or multinomial samples 3. As a test of goodness of fit
  • 73.
    www.indiande ntalacademy.c om Requirement to applychi square test Random samples Qualitative data Lowest observed frequency not less than 5 Contingency table Frequency table where sample classified according to two different attributes 2 rows ; 2 columns => 2 X 2 contingency table r rows : c columns => rXc contingency table ( ) ∑ − = E EO 2 2 χ O – observed frequency E – expected frequency
  • 74.
    www.indiande ntalacademy.c om  Steps 1. Statenull & alternate hypothesis 2. Make contingency table of the data 3. Determine expected frequency by 4. Calculate chi-square of each by- ( )cr × ( )frequencytotalN cr E × = ( ) E EO 2 2 − =χ
  • 75.
    www.indiande ntalacademy.c om 5. calculate degreeof freedom 6. Sum all the chi-square of each cell – this gives chi-square value of the data 7. Compare the calculated value with the table value at any LOS 8. Draw conclusions ( ) ∑ − = E EO 2 2 χ ( )( )11 −−= rcdf
  • 76.
    www.indiande ntalacademy.c om Example from adental health campaign School Oral hygiene Total G F+ F- P Below avg 62 (85.9) 103 (93.0) 57 (45.2) 11 (8.9) 233 Avg 50 (43.9) 36 (47.5) 26 (23.1) 7 (4.6) 119 Above avg 80 (62.3) 69 (67.5) 18 (32.8) 2 (6.5) 169 Total 192 208 101 20 521 ( )frequencytotalN cr E × = ( )( ) 63211 =×=−−= rcdf ( ) 4.31 2 2 = − = ∑ E EO cχ 22.46is0.001PatTable 2 =tχ Hence significant difference
  • 77.
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    www.indiande ntalacademy.c om Alternate formulae If wehave contingency table ( ) ( )( )( )( ) 1with 2 2 = ++++ − = df dbcadcba bcadN χ ( )( )( )( ) 1with 2 2 2 = ++++       −− = df dbcadcba N bcadN χ a b a+b c d c+d a+c b+d a+b+c+d=N If one of the value is below 5 => Yate’s correction formula
  • 79.
    www.indiande ntalacademy.c om If the tableis larger than 2X2, Yate’s correction cannot be applied – then the small frequency (<5) can be pooled or combined with next group or class in the table Chi square test only tells the presence or absence of association, but does not measure the strength of association
  • 80.
    www.indiande ntalacademy.c om  If degreeof association as to be calculated then – 1. Yule’s coefficient of association bcad bcad Q + − = ad bc ad bc Y + − = 1 1 ( )( )( )( )dcdbcaba bcad V ++++ − = 2 2 χ χ + = N C 4. Pearson’s coefficient of contingency 3. - 2. Yule’s coefficient of colligation
  • 81.
    www.indiande ntalacademy.c om Wilcoxon signed ranktest Is equivalent to paired ‘t’ test Steps Exclude any differences which are zero Put the remaining differences in ascending order, ignoring the signs Gives ranks from lowest to highest If any differences are equal, then average their ranks Count all the ranks of positive differences – T+ Count all the ranks of negative differences – T-
  • 82.
    www.indiande ntalacademy.c om If there isno differences b/n variables then T+ & T_ will be similar, but if there is difference then one sum will be large and the other will be much smaller T= smaller of T+&T_ Compare the T value with the critical value for 5%, 2% & 1% significance level A result is significant if it is smaller than critical value
  • 83.
    www.indiande ntalacademy.c om Example:Results of aplacebo-controlled clinical trail to test the effectiveness of sleeping drug Patients Sleep hrs Drug Placebo 1 6.1 5.2 2 7.0 7.9 3 8.2 3.9 4 7.6 4.7 5 6.5 5.3 6 8.4 5.4 7 6.9 4.2 8 6.7 6.1 9 7.4 3.8 10 5.8 6.3 Difference 0.9 -0.9 4.3 2.9 1.2 3.0 2.7 0.6 3.6 -0.5 Rank with signs + - 3.5 - - -3.5 10 - 7 - 5 - 8 - 6 - 2 - 9 - - -1 50.5 -4.5
  • 84.
    www.indiande ntalacademy.c om Calculated T=-4.5 df=10, Table value at 5% (n=10)= 8 Cal T< table value, H0 is rejected We conclude that sleeping drug is more effective than the placebo
  • 85.
    www.indiande ntalacademy.c om Mann Whitney Utest Is used to determine whether two independent sample have been drawn from same sample It is a alternative to student ‘t’ test & requires at least ordinal or normal measurement ( ) 21 11 21 2 1 RorR nn nnU − + += Where, n1n2 are sample sizes R1 R2 are sum of ranks assigned to I & II group
  • 86.
    www.indiande ntalacademy.c om All the observationin two samples are ranked numerically from smallest to largest without regarding the groups Procedure Then identify the observation for I and II samples Sum of ranks for I and II sample determined separately Take difference of two sum T =R1 - R2
  • 87.
    www.indiande ntalacademy.c om Comparison of birthweights of children born to 15 non smokers with those of children born to 14 heavy smokers NS 3.9 3.7 3.6 3.7 3.2 4.2 4.0 3.6 3.8 3.3 4.1 3.2 3.5 3.5 2.7 HS 3.1 2.8 2.9 3.2 3.8 3.5 3.2 2.7 3.6 3.7 3.6 2.3 2.3 3.6 R1 26 23 16 21 8 29 27 17 24 12 28 10 15 13 03 R2 7 5 6 11 25 14 9 4 20 22 19 2 1 18 Ranks assignments
  • 88.
    www.indiande ntalacademy.c om Sum of R1=272 and Sum of R2=163 Difference T=R1 – R2 is 109 The table value of T0.05 is 96 , so reject the H0 We conclude that weights of children born to the heavy smokers are significantly lower than those of the children born to the non-smokers (p<0.05)
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  • 92.
    www.indiande ntalacademy.c om Interested in relationship b/ntwo variables Are both continuous variablesUse linear regression Is one variable continuous & other categorical Use analysis of variance Or Logistic regression Both variables categorical Use contingency table analysis Two variable problem Yes No Yes No
  • 93.
    www.indiande ntalacademy.c om Research interested inrelationship B/n more than two variables Use multiple regression Or Multivariate analysis Multiple – variable problem
  • 94.
    www.indiande ntalacademy.c om Conclusion Statistics are excellenttools in research data analysis; how ever, if inappropriately used they may make the results of a well conducted research study un-interpretable or meaningless
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